*338*
*19*
*22MB*

*English*
*Pages 336*
*Year 2006*

- Author / Uploaded
- T. Makabe
- Z. Petrovic

- Similar Topics
- Technique
- Electronics

PLASMA ELECTRONICS: Applications in Microelectronic Device Fabrication

© 2006 by Taylor & Francis Group, LLC

Series in Plasma Physics Series Editor:

Steve Cowley, Imperial College, UK and UCLA, USA An Introduction to Inertial Coninement Fusion S Pfalzner Other books in the series:

Aspects of Anomalous Transport in Plasmas R Balescu Non-Equilibrium Air Plasmas at Atmospheric Pressure K H Becker, R J Barker and K H Schoenbach (Eds) Magnetohydrodynamic Waves in Geospace: The Theory of ULF Waves and their Interaction with Energetic Particles in the Solar-Terrestrial Environment A D M Walker Plasma Physics via Computer Simulation (paperback edition) C K Birdsall, A B Langdon Plasma Waves, Second Edition D G Swanson Microscopic Dynamics of Plasmas and Chaos Y Elskens and D Escande Plasma and Fluid Turbulence: Theory and Modelling A Yoshizawa, S-I Itoh and K Itoh The Interaction of High-Power Lasers with Plasmas S Eliezer Introduction to Dusty Plasma Physics P K Shukla and A A Mamun The Theory of Photon Acceleration J T Mendonça Laser Aided Diagnostics of Plasmas and Gases K Muraoka and M Maeda Reaction-Diffusion Problems in the Physics of Hot Plasmas H Wilhelmsson and E Lazzaro The Plasma Boundary of Magnetic Fusion Devices P C Strangeby Non-Linear Instabilities in Plasmas and Hydrodynamics S S Moiseev, V N Oraevsky and V G Pungin

© 2006 by Taylor & Francis Group, LLC

Series in Plasma Physics

PLASMA ELECTRONICS: Applications in Microelectronic Device Fabrication

T Makabe Keio University, Japan

Z Petrovic´ Institute of Physics Belgrade, Serbia

New York London

© 2006 by Taylor & Francis Group, LLC

Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-7503-0976-8 (Hardcover) International Standard Book Number-13: 978-0-7503-0976-9 (Hardcover) Library of Congress Card Number 2005056888 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data Makabe, T. (Toshiaki). Plasma electronics : applications in microelectronic device fabrication / by T. Makabe and I. Petrovic p. cm. -- (Series in plasma physics) Includes bibliographical references and index. ISBN 0-7503-0976-8 1. Plasma engineering. I. Title. II. Series. TA2020.M35 2005 621.044--dc22

2005056888

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc.

© 2006 by Taylor & Francis Group, LLC

and the CRC Press Web site at http://www.crcpress.com

Preface

Over the past three decades low-temperature plasma applications have been extended from primarily lighting to the fabrication of microelectronic devices and new materials, far exceeding our expectations. Radio frequency plasmas ranging from 105 Hz to 109 Hz, in particular, are now used to process metallic, semiconductor, and dielectric materials for the fabrication of ultra largescale integrated (ULSI) circuits and to deposit various kinds of functional thin films and to modify the surface properties. Without plasma-produced ions and dissociated neutral radicals for etching and deposition on wafers, microelectronics manufacturing for ULSI circuit would simply be unfeasible. The advent of ULSI fabrication has greatly changed how the field of plasma science is approached and understood. Low-temperature non-equilibrium plasmas are sustained mainly by electron impact ionization of a feed gas driven by an external radio frequency power source. These low-temperature plasmas acquire characteristics intrinsic to the feed gas molecules as determined by their unique electron-collision cross-section sets. This uniqueness means that plasmas must be understood using quantum, atomic, and molecular physics. The disparate time and spatial scales involved in low-temperature plasma processing (submeter to nanometer and seconds to nanoseconds) makes plasma processing an inherently stiff problem. The characteristics of low-temperature plasmas contrast markedly with highly ionized equilibrium plasmas that are ensembles of charged particles whose behavior can be understood through their long-range Coulomb interactions and collective effects and characterized by plasma frequency, Debye length, and electron temperature. The fundamental collision and reaction processes occurring both in gas phase and on surface in low-temperature plasmas are the bases for understanding their behavior and exploiting them for practical applications. In the emerging nanotechnology era, device design, reliability, and the design of integrated plasma processes for their fabrication are tightly coupled. Being able to predict feature profile evolution under the influence of spatio-temporally varying plasmas is indispensable for the progress of nanotechnology. Prediction of plasma damage and its mitigation are also crucial prediction & mitigation are 2 process and will be performed through a series of vertically integrated numerical modeling and simulations ranging from the reactor scale to those cognizant of device elements subtending the plasma. Motivated by the important role of plasmas in technology and the need for simulations to understand the associated complex processes, we emphasize

© 2006 by Taylor & Francis Group, LLC

in this book academic fusion among atomic and molecular physics, surface physics, the Boltzmann transport theory, electromagnetic theory, and computational science as plasma electronics. We do this to describe and predict the space and time characteristics of low-temperature plasmas and associated processing intrinsic to specific feed gases. An underlying theme through this work is computer-aided plasma analysis and synthesis, with emphasis on computational algorithms and techniques. This book is based on a series of lectures presented at Keio University as part of its graduate program. The university’s interest in the subject matter and feedback were essential parts of developing this text. We believe that the book is well suited as an instrument for self-instruction through its topical exercises and problems arranged in each chapter. It is a pleasure to acknowledge our debt to David Graves and Robert Robson, who have helped in a variety of ways during the long period of our research life composing plasma electronics. Finally, we are indebted to T. Yagisawa for figure preparation and his attention to detail. Toshiaki Makabe and Zoran Lj. Petrovic Keio University

© 2006 by Taylor & Francis Group, LLC

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Plasma and Its Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Application of Low-Temperature Plasma . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Academic Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Phenomenological Description of the Charged Particle Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Transport in Real (Configuration) Space . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Momentum Balance of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Energy Balance of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Transport in Velocity Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Electron Velocity Distribution and Swarm Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Ion Velocity Distribution and Mean Energy . . . . . . . . . . . . . . . 18 2.3 Thermal Equilibrium and Its Governing Relations . . . . . . . . . . . . . . . 20 2.3.1 Boltzmann Distribution in Real Space . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Maxwell Distribution in Velocity Space . . . . . . . . . . . . . . . . . . . 22 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Macroscopic Plasma Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Quasi-Neutrality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 3.3 Charge-Separation in Plasmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 3.3.1 Spatial Scale of Charge-Separation . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.2 Time Scale for Charge-Separation . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Plasma Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.1 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.2 Metal Probe in a Plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 3.5 Particle Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5.1 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5.2 Spatial and Time Scale of Diffusion . . . . . . . . . . . . . . . . . . . . . . . 35 3.6 Bohm Sheath Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6.1 Bohm Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6.2 Floating Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

© 2006 by Taylor & Francis Group, LLC

4 Elementary Processes in Gas Phase and on Surfaces. . . . . . . . . . . . . . . . .41 4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

Particles and Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.1 Particle Representation in Classical and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.2 Locally Isolated Particle Group and Wave Packets . . . . . . . . 44 Collisions and Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.1 Conservation Laws in Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2.2 Definition of Collision Cross Sections. . . . . . . . . . . . . . . . . . . . .49 4.2.3 The Distribution of Free Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.4 Representation of Collisions in Laboratory and CM Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Classical Collision Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 4.3.1 Scattering in Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.2 Conditions for the Applicability of the Classical Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Quantum Theory of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.1 Differential Scattering Cross Section σ (θ) . . . . . . . . . . . . . . . . . 66 4.4.2 Modified Effective Range Theory in Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Collisions between Electrons and Neutral Atoms/Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5.1 Resonant Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Electron–Atom Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.6.1 Energy Levels of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.6.2 Electron–Atom Scattering Cross Sections . . . . . . . . . . . . . . . . . 77 Electron–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.7.1 Rotational, Vibrational, and Electronic Energy Levels of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.7.2 Rotational Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7.2.1 Rotational Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7.2.2 Rotational Excitation Cross Sections . . . . . . . . . . . . . 84 4.7.3 Vibrational Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.7.3.1 Vibrational Energy Levels . . . . . . . . . . . . . . . . . . . . . . . 87 4.7.3.2 Vibrational Cross Sections . . . . . . . . . . . . . . . . . . . . . . . 88 4.7.4 Electronic Excitation and Dissociation . . . . . . . . . . . . . . . . . . . . 89 4.7.4.1 Electronic States of Molecules. . . . . . . . . . . . . . . . . . . .89 4.7.4.2 Cross Sections for Electronic Excitation of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7.5 Electron Collisions with Excited Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Nonconservative Collisions of Electrons with Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.8.1 Electron-Induced Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.8.2 Electron Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.8.2.1 Dissociative Electron Attachment . . . . . . . . . . . . . . . . 99 4.8.2.2 Nondissociative Electron Attachment . . . . . . . . . . . 100

© 2006 by Taylor & Francis Group, LLC

4.8.2.3 Ion Pair Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.8.2.4 Electron Attachment to Excited Molecules . . . . . 101 4.8.2.5 Rate Coefficients for Attachment . . . . . . . . . . . . . . 102 4.8.3 Electron–Ion and Ion–Ion Recombination . . . . . . . . . . . . . . 103 4.8.4 Electron–Ion and Electron–Electron Collisions. . . . . . . . . .105 4.9 Heavy Particle Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.9.1 Ion–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.9.1.1 Charge Transfer, Elastic, and Inelastic Scattering of Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.9.1.2 Ion–Molecule Reactions . . . . . . . . . . . . . . . . . . . . . . . 109 4.9.2 Collisions of Fast Neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.9.3 Collisions of Excited Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.9.3.1 Chemi-Ionization and Penning Ionization . . . . . 112 4.9.4 Collisions of Slow Neutrals and Rate Coefficients . . . . . . 116 4.9.4.1 Quenching and Transport of Excited States . . . . 116 4.9.4.2 Kinetics of Rotational and Vibrational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.10 Photons in Ionized Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.10.1 Emission and Absorption of Line Radiation . . . . . . . . . . . 119 4.10.2 Resonant Radiation Trapping . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.11 Elementary Processes at Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.11.1 Energy Levels of Electrons in Solids . . . . . . . . . . . . . . . . . . . 124 4.11.2 Emission of Electrons from Surfaces. . . . . . . . . . . . . . . . . . .127 4.11.2.1 Photo-Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.11.2.2 Thermionic Emission . . . . . . . . . . . . . . . . . . . . . . . 128 4.11.2.3 Field-Induced Emission . . . . . . . . . . . . . . . . . . . . . 129 4.11.2.4 Potential Ejection of Electrons from Surfaces by Ions and Excited Atoms . . . . . . . . . . . . . . . . . . 132 4.11.3 Emission of Ions and Neutrals from Surfaces . . . . . . . . . . 135 4.11.3.1 Surface Neutralization . . . . . . . . . . . . . . . . . . . . . . 136 4.11.3.2 Surface Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.11.4 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5 The Boltzmann Equation and Transport Equations of Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2.1 Transport in Phase Space and Derivation of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.3 Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4 The Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.4.1 Conservation of Number Density . . . . . . . . . . . . . . . . . . . . . . 151 5.4.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.4.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.5 Collision Term in the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 153

© 2006 by Taylor & Francis Group, LLC

5.5.1 Collision Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.5.2 Collision Integral between an Electron and a Gas Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.5.2.1 Elastic Collision Term J elas . . . . . . . . . . . . . . . . . . . . . . 155 5.5.2.2 Excitation Collision Term J ex . . . . . . . . . . . . . . . . . . . 158 5.5.2.3 Ionization Collision Term J ion . . . . . . . . . . . . . . . . . . . 158 5.5.2.4 Electron Attachment Collision Term J att . . . . . . . . 159 5.6 Boltzmann Equation for Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.6.1 Spherical Harmonics and Their Properties . . . . . . . . . . . . . . 160 5.6.2 Velocity Distribution of Electrons . . . . . . . . . . . . . . . . . . . . . . . 163 5.6.2.1 Velocity Distribution under Uniform Number Density: g 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.6.2.2 Velocity Distribution Proportional to ∇r n(r, t): g 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 5.6.3 Electron Transport Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6 General Properties of Charged Particle Transport in Gases . . . . . . . . . 175 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2 Electron Transport in DC-Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2.1 Electron Drift Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.2.2 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.2.3 Mean Energy of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.2.4 Excitation, Ionization, and Electron Attachment Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.3 Electron Transport in Radio-Frequency Electric Fields . . . . . . . . . . 184 6.3.1 Relaxation Time Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.3.2 Effective Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.3 Expansion Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.3.4 Direct Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.3.5 Time-Varying Swarm Parameters . . . . . . . . . . . . . . . . . . . . . . . 199 6.4 Ion Transport in DC-Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

7 Modeling of Nonequilibrium (Low-Temperature) Plasmas. . . . . . . . .205 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.2 Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.2.1 Governing Equations of a Continuum Model . . . . . . . . . . . . 207 7.2.2 Local Field Approximation (LFA) . . . . . . . . . . . . . . . . . . . . . . . 210 7.2.3 Quasi-Thermal Equilibrium (QTE) Model . . . . . . . . . . . . . . . 211 7.2.4 Relaxation Continuum (RCT) Model . . . . . . . . . . . . . . . . . . . . 212 7.2.5 Phase Space Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 7.3 Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3.1 Monte Carlo Simulations (MCSs) . . . . . . . . . . . . . . . . . . . . . . . 216 7.3.2 Particle-in-Cell (PIC) and Particle-in-Cell/Monte Carlo Simulation (PIC/MCS) Models . . . . . . . . . . . . . . . . . . . . . . . . . 219

© 2006 by Taylor & Francis Group, LLC

7.4 7.5

Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Circuit Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221 7.5.1 Equivalent Circuit Model in CCP . . . . . . . . . . . . . . . . . . . . . . . 221 7.5.2 Equivalent Circuit Model in ICP . . . . . . . . . . . . . . . . . . . . . . . . 223 7.5.3 Transmission-Line Model (TLM) . . . . . . . . . . . . . . . . . . . . . . . . 224 7.6 Electromagnetic Fields and Maxwell’s Equations . . . . . . . . . . . . . . . 225 7.6.1 Coulomb’s Law, Gauss’s Law, and Poisson’s Equation . . . 225 7.6.2 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.6.3 Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.6.4 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8 Numerical Procedure of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1

Time Constant of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1.1 Collision-Oriented Relaxation Time . . . . . . . . . . . . . . . . . . . . . 231 8.1.2 Plasma Species-Oriented Time Constant. . . . . . . . . . . . . . . . .232 8.1.3 Plasma-Oriented Time Constant/Dielectric Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.2 Numerical Techniques to Solve the Time-Dependent Drift-Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.2.1 Time-Evolution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.2.1.1 Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.2.1.2 Digitalization and Stabilization . . . . . . . . . . . . . . . . . 237 8.2.1.3 Time Discretization and Accuracy . . . . . . . . . . . . . . 239 8.2.2 Scharfetter–Gummel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.2.3 Cubic Interpolated Pseudoparticle Method . . . . . . . . . . . . . . 243 8.2.4 Semi-Implicit Method for Solving Poisson’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245 8.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.3.1 Ideal Boundary — Without Surface Interactions . . . . . . . . . 246 8.3.1.1 Dirichlet Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246 8.3.1.2 Neumann Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.3.1.3 Periodicity Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.3.2 Electrode Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.3.2.1 Metallic Electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.3.2.2 Dielectric Electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . .249 8.3.3 Boundary Conditions with Charge Exchange . . . . . . . . . . . . 250 8.3.4 Boundary Conditions with Mass Transport . . . . . . . . . . . . . . 250 8.3.4.1 Plasma Etching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 8.3.4.2 Plasma Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.4.3 Plasma Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.5 Moving Boundary under Processing . . . . . . . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

9

Capacitively Coupled Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.1 Radio-Frequency Capacitive Coupling . . . . . . . . . . . . . . . . . . . . . . . . 255

© 2006 by Taylor & Francis Group, LLC

9.2

Mechanism of Plasma Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.2.1 Low-Frequency Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.2.2 High-Frequency Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 9.2.3 Electronegative Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 9.2.4 Very High-Frequency Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.2.5 Two-Frequency Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 9.2.6 Pulsed Two-Frequency Plasma . . . . . . . . . . . . . . . . . . . . . . . . . 271 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

10 Inductively Coupled Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.1 10.2

Radio-Frequency Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . . 275 Mechanism of Plasma Maintenance. . . . . . . . . . . . . . . . . . . . . . . . . .275 10.2.1 E-mode and H-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.2.2 Mechanism of Plasma Maintenance . . . . . . . . . . . . . . . . . . 277 10.2.3 Effect of Metastables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10.2.4 Function of ICP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.3 Wave Propagation in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.3.1 Plasma and Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.3.2 ICP and the Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

11 Magnetically Enhanced Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.1 Direct-Current Magnetron Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 11.2 Unbalanced Magnetron Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 11.3 Radio-Frequency Magnetron Plasma. . . . . . . . . . . . . . . . . . . . . . . . .294 11.4 Magnetic Confinements of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . 295 11.5 Magnetically Resonant Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

12 Plasma Processing and Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 12.1 12.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Physical Sputtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .301 12.2.1 Target Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 12.2.2 Sputtered Particle Transport . . . . . . . . . . . . . . . . . . . . . . . . . 306 12.3 Plasma Chemical Vapor Deposition . . . . . . . . . . . . . . . . . . . . . . . . . .308 12.3.1 Plasma CVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 12.3.2 Large-Area Deposition with High Rate . . . . . . . . . . . . . . 311 12.4 Plasma Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 12.4.1 Wafer Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 12.4.1.1 On Electrically Isolated Wafers (without Radio-Frequency Bias) . . . . . . . . . . . . . . . . . . . . . 313 12.4.1.2 On Wafers with Radio-Frequency Bias . . . . . 313 12.4.2 Selection of Feed Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 12.4.3 Si or Poly-Si Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 12.4.4 Al Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

© 2006 by Taylor & Francis Group, LLC

12.4.5 SiO2 Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 12.4.6 Feature Profile Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 12.4.7 Plasma Bosch Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 12.4.8 Charging Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 12.4.8.1 Surface Continuity and Conductivity . . . . . . 326 12.4.8.2 Charging Damage to Lower Thin Elements in ULSI. . . . . . . . . . . . . . . . . . . . . . . . . .328 12.4.9 Thermal Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

© 2006 by Taylor & Francis Group, LLC

1 Introduction

1.1 Plasma and Its Classification The plasma state is defined as the fourth state of matter as distinct from the solid, liquid, and gas phases. It consists of free positive and negative charges with electrical quasi-neutrality in addition to feed gas components. The word “plasma” was first introduced by Langmuir in 1928 and derives from the Greek πλασµα meaning “something formed” [1]. The strength of the plasma state is described in terms of the ionization degree HDI = n p /N, where ne (= n p ) and N are the number density of the charged particles and neutral molecules in a plasma. Strongly ionized plasma is realized in HDI ≥ 10−3 . In particular, electron transport in a strongly ionized plasma is subject to the long-range Coulomb interaction with surrounding electrons and positive ions, and the smooth trajectory makes it impossible to identify changes in momentum by collision during flight (Figure 1.1a). Therefore, strongly ionized plasma is termed collisionless plasma. On the other hand, the characteristics of a weakly ionized plasma are properly represented by the shortrange interaction between the electrons and the neutral molecules in the feed gas under external electric or magnetic fields. In such a collision-dominated plasma, the electron trajectory is characterized by both flight in the field and collision with surrounding molecules (Figure 1.1b). In comparison, a neutral molecule has a straight trajectory under a short-range collision in gases (Figure 1.1c). In the gas phase, a plasma state can be produced electrically, thermally, or optically through the ionization of neutral molecules in the feed gas. In particular, nonequilibrium plasma or low-temperature plasma, which is produced by the collisional ionization of free electrons under an external electrical power, has a property that the electron energy is much higher than that of the neutral gas. We herein distinguish low-temperature plasma in the generation mechanism from thermal plasma produced by the thermal ionization of neutral molecules. Low-temperature plasmas from micrometer to meter size are artificially maintained even in an electrodeless reactor as well as between metallic or dielectric electrodes by using electrical power sources ranging from direct current to radiofrequency levels (tens of kHz to several hundred MHz and GHz). 1 © 2006 by Taylor & Francis Group, LLC

2

Plasma Electronics: Applications in Microelectronic Device Fabrication (a)

(b)

(c) E

FIGURE 1.1 Electron trajectory in three different media. In strongly ionized plasma (a), in weakly ionized plasma (b), and in neutral gas (c).

1.2 Application of Low-Temperature Plasma Low-temperature plasma technology differs from that of collisionless plasma in that a low-temperature plasma is produced and maintained in a collisiondominated region and exhibits proper characteristics and functions intrinsic to the quantum state of the feed gas molecules. This is one of the primary advantages of low-temperature plasmas for material processing and device fabrication requiring a variety of surface processes and very different reactions among materials adjacent to each other. The technology assisted by low-temperature plasma is generally referred to as plasma processing and is classified into plasma-enhanced chemical vapor deposition, plasma etching, sputtering, ashing, surface modification, and so on, on a scale of size ranging from nanometers to meters (Figure 1.2). Typical geometrical arrangements of Pressure (Pa) 1013

Plasma density (cm-3)

101

1.0

1012 ICP ECR etching 1011 doping 1010 10

CCP

102

103

ICP MEMS ashing trimming

ICP surface modification

Barrierdischarge

108 107

105

µ -plasma PDP

Magnetron film deposition sputterCCP deposition

9

104

10-3

10-2

10-1

1.0

10-3

Pressure (Torr) FIGURE 1.2 Low-temperature plasma and material processes.

© 2006 by Taylor & Francis Group, LLC

102

103

Introduction

3

Passivation

M6

M5

M4

Global interconnect (up to five layers) Via

M3

Cu

Low-k

Intermediate interconnect (up to four layers)

M2

Local interconnect

M1 SiO2

n+ p-well

n+

STI

nMOS

p+

p+

pMOS

n-well

substrate

FIGURE 1.3 Typical large scale integrated circuits.

an ultra-large-scale integrated circuit demonstrated partly in Figure 1.3 are manufactured through more than 100 plasma processes. The principal plasma processes for material fabrication are etching and deposition, which are highly competitive. In an era of nanotechnology, plasma technology has developed into a combination of top-down and bottom-up processes. The bottom-up (chemical) approach is effectively achieved under plasma-enhanced conditions.

1.3 Academic Fusion In the past two decades, great progress has been made in the application of low-temperature plasma [2]. Most currently used models of low-temperature radiofrequency plasma were proposed between the mid-1980s and the early

© 2006 by Taylor & Francis Group, LLC

4

Plasma Electronics: Applications in Microelectronic Device Fabrication

Low-Temperature Plasma and Its Technologies

FIGURE 1.4 Academic fusion for low-temperature plasmas.

1990s [3]. Further achievements in plasma technology at the design level will lead to greater miniaturization of ultra-large-scale integrated (ULSI) circuits in microelectronics and nanoelectronics and to the further functionalization of new materials. These developments will be synchronized with the progress of computers utilizing high-speed and high-performance ULSI chips. We stand at the advent of the design era of sophisticated plasmas based on atomic and molecular physics both in the gas phase and in the surface phase. The low-temperature plasma and related technologies will join the exciting and practical fields of academic fusion reconstructed by computational science with quantum atomic and molecular physics, surface physics and chemistry, Boltzmann equation of particles, and Maxwell’s electromagnetic theory as is shown in Figure 1.4. To address the design issue, it is essential to prepare a two- or threedimensional image of the plasma structure in a reactor and feature profile on a substrate surface by experimental observations or computational modeling.

References 1. Goldston R.J. and Rutherford P.H. 1995. Introduction to Plasma Physics. Bristol: IOP. 2. American Institute of Physics. 2003. 50 years of science, technology, and the AVS (1953–2003), J. Vac. Sci. Technol. A (Special Issue) 21:5. 3. Makabe, T., Ed. 2002. Advances in Low Temperature RF Plasmas, Basis for Process Design. Amsterdam: Elsevier.

© 2006 by Taylor & Francis Group, LLC

2 Phenomenological Description of the Charged Particle Transport

In this chapter we attempt to lay out the foundation of the phenomenology of charged particle transport in gases. We are primarily concerned with electrons, although ions are also covered briefly. We leave the more detailed description of transport theory for one of the later chapters; here we describe aspects of transport that may be understood from a phenomenological consideration of the charged particles either in the real or in velocity space. We derive the corresponding velocity distributions and transport (swarm) properties and some of their basic relationships. We also derive general equilibrium distributions of charged particles in real and velocity space in thermal equilibrium.

2.1 Transport in Real (Configuration) Space Typically, without the external electric field, a charged particle (we discuss mainly the case of electrons here) swarm will acquire a Gaussian spatial profile in density. This profile is the result of the particle’s random thermal motion and collisions with gas molecules, provided that initially all particles start from the same position. Long-range Coulomb interactions between charged particles have no influence on the development of the swarm in a weakly ionized gas. As a result, electrons will have a net movement in the negative direction of the field superimposed on their random motion by collisions in all directions under an external field. When a field is added, the Gaussian spatial profile will be slightly shifted and even skewed in the direction opposite to that of the field. A swarm of electrons and its development are considered in a direct current (DC) electric field E(= −Ek), where k is a unit vector along the z-axis. For the purpose of this chapter, we assume that the transport properties are uniformly distributed throughout the electron swarm, and we mainly follow the semiquantitative analysis of Parker and Lowke [1].

5 © 2006 by Taylor & Francis Group, LLC

6

Plasma Electronics: Applications in Microelectronic Device Fabrication n(r)

E

I

II

z

III

0 r

FIGURE 2.1 Spatial profile of electron density for a swarm released from a single point in space and allowed to develop under the influence of electric field and collisions with background molecules.

2.1.1 Momentum Balance of Electrons The effect of field will also result in the net flux of electrons (Γ) in the direction opposite to the field. In other words, a net effective velocity (v0 ) may be used to describe the average motion of a swarm of electrons with number density distribution n(r). If a swarm introduced into an electrical field acquires a uniform distribution, it will attain an average, drift velocity (vd ). When we release a swarm of particles from a single point as described above, then there will be a spatial density distribution of charged particles n(r), as shown in Figure 2.1. Thus the density gradient will result in flux due to diffusion (−D∂n/∂r) in the direction opposite to that of the gradient. The total flux of particles in one-dimensional space along a DC electric field E = −Ek is then given by Γ = nv0 = nvd − D

∂n k, ∂z

(2.1)

where D is the diffusion coefficient. The momentum balance of electrons will be applied to derive the transport parameters. Electrons gain velocity and consequently average momentum only along the axis of the electrical field. Then, the total change of the momentum of the swarm per unit time is equal to the impulse eE. The momentum change is caused by the collision with background gas, mainly the elastic collision with the rate of Rm in a low-energy electron. The momentum balance equation is therefore written as D ∂n d k Rm , (mnvz k) = e(−E)n − mn vd − dt n ∂z

(2.2)

where m and e(> 0) are the mass and charge of the electron, respectively. In the stationary state, dvz /dt = 0, the total differential on the left-hand side of Equation 2.2 is separated into two parts, one of which is zero. Applying

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

7

transformation vz = ∂z/∂t, we separate the Equation 2.2 is into two with respect to the electron number density n(r) and its spatial derivative 0 = (−eE − mvd Rm ) n, ∂n 0 = mvz2 − mDRm . ∂z

(2.3)

It is justified (as shown later) to assume that the z component of the velocity is not substantially perturbed by the electric field-induced drift, and therefore 13 v 2 = vx2 = v 2y ∼ vz2 . In that case we may simplify the second part of Equation 2.3 to give the approximate value of the diffusion coefficient as vd = − D=

eE . mRm

(2.4)

v2 . 3Rm

(2.5)

PROBLEM 2.1.1 Derive Equations 2.4 and 2.5. 2.1.2 Energy Balance of Electrons The former derivation of the properties of the electron swarm relied only on momentum balance as given by Equation 2.2. Now, we consider the energy balance starting from the concept of the mean energy εm . The change of the total energy of the system of electrons with n(r) is given by d 2m D ∂n k −n εm Rm , (2.6) (nεm ) = e(−E)n · vd − dt n ∂z M where M is the mass of the gas molecules. The first term on the right-hand side represents the energy gain by the movement along the field axis in unit time, and the second term represents the collisional energy losses, given by the sum of the elastic loss, 2mεm /M. When we substitute Equations 2.4 and 2.5 in the energy balance of Equation 2.6, we obtain for the stationary state 2e Eεm 1 ∂n 2m e2 E2 − − εm Rm = 0. mRm 3mRm n ∂z M

(2.7)

We consider solving this equation under certain conditions, particularly in regard to the spatial profile of the electron number density n(z). 1. ∂n/∂z = 0: This corresponds to the point I in Figure 2.1, that is, to the maximum of the density distribution. From Equation 2.7 we then obtain M eE 2 . (2.8) = εm0 = εm | ∂n ∂z =0 2m2 Rm

© 2006 by Taylor & Francis Group, LLC

8

Plasma Electronics: Applications in Microelectronic Device Fabrication 2. ∂n/∂z = 0: This corresponds to all points except for I in Figure 2.1. For the region with a negative density gradient (region II in Figure 2.1) the mean energy is higher than that at the peak (point I): εm = εm0 + ε.

So far we have assumed that the collisional rate was independent of the electron energy. However, this is normally not the case. Thus we use a Taylor expansion around the mean energy at the peak I for the collision rate: Rm (εm ) = Rm0 +

∂ Rm |0 ε, ∂ε

which in combination with Equations 2.7 gives 2ε 1 ∂n m0 . εm = εm0 1 − Rm εm0 n ∂z 3eE 1 + 2 ∂∂ε |0 Rm0

(2.9)

Exercise 2.1.1 Discuss the value of the mean electron energy in two cases, (a) ∂n/∂z < 0 and (b) ∂n/∂z > 0, as compared with the mean energy at the peak of the swarm εm0 . For case (a), the gradient of the electron number density shown in Figure 2.1 is negative as in region II, and therefore εm > εm0 . However, for case (b), the gradient of the electron number density is positive (see region III in Figure 2.1) and therefore εm < εm0 . Thus, unless the collision rate Rm changes very rapidly, we may expect the mean energy of electrons to increase toward the front of the swarm where flux due to diffusion adds to the driftinduced component. The spatial profile of energy may have consequences on the transport coefficients, provided that some energy-dependent processes also exist, which is generally the case. PROBLEM 2.1.2 Derive expression 2.9 for the mean electron energy εm . In the presence of density gradients in the swarm, random motion under collisions with gas molecules will result in diffusion. So far we have considered diffusion only in one dimension along the direction of the electric field. When, however, we consider three-dimensional space defined by the Cartesian coordinates r(x, y, z) and the corresponding unit vectors (i, j, k), while keeping the same direction of the electric field E = −Ek, the electron number density develops as n(x, y, z). The conditions of the zero gradient in one direction now become lines along the surface of the density profile. The total flux of electrons in Equation 2.1 are revised for the three-dimensional case as ∂n ∂n ξ ∂n Γ = nvd k − D0 i+ j − D0 1 − k, (2.10) ∂x ∂y 1 + 2ξ ∂z

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

9

where εm0 ∂Rm ξ= 0. Rm0 ∂ε

(2.11)

As a result of the influence of the electric field, the diffusion along the z-axis (direction of the field) has special properties and is in general different from the diffusion in perpendicular directions. The diffusion of electrons (and charged particles in general) becomes anisotropic with two components of the diffusion tensor D: DL (longitudinal diffusion coefficient) and DT (transverse diffusion coefficient). These two coefficients may be given as DL = D0 1 −

ξ 1 + 2ξ

DT = D0 =

v2 , 3Rm

,

(2.12)

and (2.13)

which are valid only for approximations that were involved in the derivation of the formulae. In the case where the collision rate Rm is independent of the mean energy, the diffusion will be isotropic. The same is true for the zero electric field. In this section we have seen that the transport of electrons (the same is true for ions) may be described with the aid of transport coefficients. We have given approximate relations for the drift velocity and components of the diffusion tensor. In general these properties will depend on the mean electron energy. However, it is not practical to use mean energy for tabulating the data for transport coefficients. It is better to use an external parameter associated with the value of the external electric field E. The reduced electric field is defined as the ratio between the magnitude of the electric field E and the neutral gas number density N, that is, E/N. It is shown that the reduced field is proportional to the energy gained by electrons between two collisions. E/N is expressed in units known as Townsend (in honor of the founder of gaseous electronics), which is defined as 1 Td = 10−21 Vm2 = 10−17 Vcm2 . PROBLEM 2.1.3 By substituting Equation 2.9 into Equation 2.6, derive the longitudinal DL (Equation 2.12) and transverse DT (Equation 2.13) coefficients. Assuming analytic dependence of the collision rate Rm on energy find the ratio DL /DT in analytic form. Discuss the conditions when the ratio is equal to, less than, or greater than 1. (A hint: choosing Rm ∼ (εm /εm0 )(ℓ+1)/2 is particularly convenient.)

© 2006 by Taylor & Francis Group, LLC

10

Plasma Electronics: Applications in Microelectronic Device Fabrication

PROBLEM 2.1.4 Pressure p may be used to represent the gas number density N; find the relationship between E/N and E/ p given that E/N[Vcm2 ] = 1.036 × 10−19 Tg [K]E/ p[Vcm−1 Torr−1 ], where Tg is gas temperature. Here, at Tg = 273 K, E/P of 1 Vcm−1 Torr is equal to 2.823 Td (= 10−17 Vcm2 ). Prove that the mean energy gain between two collisions is proportional to E/N.

2.2 Transport in Velocity Space An ensemble of electrons may be studied in velocity space. Whereas in the real space the spatial density distribution n(r) was used to describe the swarm, in velocity space we may consider the probability distribution of velocities. The velocity distribution g(v) is defined so that the quantity g(v)dv is the probability of finding particles with velocity v within the small element dv. Due to the symmetry with respect to the vz -axis, it is advantageous to use a polar coordinate system (v, θ, ϕ). Thus dv = v 2 sin θ dθ dϕ. The magnitude of the velocity v = |v| is associated with the energy of electrons ε = mv 2 /2, and in the case of thermal equilibrium the distribution is given by the so-called Maxwellian distribution:

g M (v) =

m 3/2 mv 2 exp − . 2π kT 2kT

(2.14)

At the same time, we may define the velocity distribution function g(v, r, t), which describes swarm development with respect to both velocity and real space and with respect to time. 2.2.1 Electron Velocity Distribution and Swarm Parameters Thermal equilibrium, however, does not occur for low-density swarms of charged particles in an electric field. Normally, an electric field affects electrons, which gain energy but are not able to dissipate it to give translational motion to gas molecules or ions, because elastic collisions are not particularly effective in energy transfer to the target due to the very different masses; that is, m/M ≪ 1 . Thus electrons in swarms and in low-density collisional plasmas are normally not in thermal equilibrium. Two distinct conditions may occur for the electron swarm in an electric field. In the case that the properties of the swarm are uniform in the real space, we will have a quasiequilibrium. Then, the energy gained by the field is dissipated in collisions, both elastic and inelastic. As a result, properties of electrons given by electron energy distribution do not change within the swarm. In such a case (i.e., under

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

11

g( )

E

vr

0

vz

FIGURE 2.2 Typical electron velocity distribution g(v) as a function of axial vz and radial vr velocities under a DC electric field.

hydrodynamic conditions), it will be possible to separate spatial and velocity distributions. However, close to the initial stage or boundary wall or soon after the release the properties of the electrons may not be uniform in space or time or both. These nonequilibrium conditions are not expressed in terms of thermodynamic equilibrium; however, in relation to the balance between energy gain and loss, the swarm will develop until the balance is achieved or the swarm is lost. Such cases are described as a nonlocal (in space) or not relaxed electron swarm transport. In this section we focus on relaxed spatially uniform swarms that are in the hydrodynamic regime. Thus the velocity distribution provides a complete description of the swarm. Figure 2.2 shows a velocity distribution in velocity space. Here, we take advantage of the azimuthal symmetry of the swarm in velocity space and present the distribution as a function of the axial (axis of electric field) velocity vz and radial (perpendicular) velocity vr . The electric field produces asymmetry of the distribution with a peak shifted in one direction. The small difference between the components in the direction and opposite to the direction of the field produces a net drift of particles. At the same time the velocities directed in opposite directions occur at all points throughout the swarm profile. In the bulk of the spatial swarm profile, loss of particles moving outside a small section is compensated by gain from the nearby sections that have a similar density; this, however, is not the case at the edges. The velocity distribution thus contains information about both drift and diffusion. PROBLEM 2.2.1 The drift velocity and mean energy of electrons in argon under the uniform field, E = 100 Vcm−1 , and at pressure of 1 Torr (at temperature of 300 K) are vd = 2.3×107 cms−1 and 8 eV, respectively. Show, then, that the mean random velocity vr and drift velocity vd satisfy the relation (vd /vr ) ≪ 1.

© 2006 by Taylor & Francis Group, LLC

12

Plasma Electronics: Applications in Microelectronic Device Fabrication

E

vr v

θ 0

0' ∆V

v'

vz

FIGURE 2.3 Contour plot of electron velocity distribution g(v) as a function of axial vz and radial vr velocities. When the electric field is turned on, a small component of velocity V is added to all electrons in the direction opposite to the field E.

In Figure 2.3 we show the effect of the electric field on the velocity distribution g(v) at each velocity v with and without the field. When the electric field E is turned on, a small component of velocity V is added to all electrons in the direction opposite to the field. In the simplified model the circular contour representing random (thermal) isotropic motion of electrons is slightly shifted as if its center were moved to the position O′ from O. The resulting velocity is v′ , which is given by: v′ − v ∼ = V cos θ (where θ is the angle between v′ and −E). We can now estimate the effect of the electric field on the electron velocity distribution as g(v′ ) = g(v + V cos θ) ∼ = g0 (v) + V cos θ

dg0 (v) , dv

(2.15)

where we applied the condition | v | ≫ | Vcos θ |. The distribution g(v′ ) is composed of an isotropic part in all directions and having, in general, a nonMaxwellian term and a small anisotropic term. Both terms are influenced by the electric field. This expansion is analogous to a two-term expansion in spherical harmonics of the velocity distribution, where the second term is the anisotropic term (see Chapter 5). Now we consider the momentum conservation of electrons with velocity distribution g(v) in gases under an electric field. First, let us define Qm (v) as the momentum transfer cross section of the electron by elastic scattering with the background gas molecule with number density N. Then, for one group of electrons with velocity v, the collision frequency of the momentum exchange is NQm (v)v, and the total momentum dissipated in collisions in the direction opposite to the field E is mv cos θ NQm (v)v. The total momentum exchange, therefore, is obtained by averaging over all velocities by using the

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

13

distribution, and in the steady state we have ∂(nm < v >) = e(−E)n − ∂t

nmvNQm (v)vg(v)dv = 0.

(2.16)

Here, g(v) is normalized as g(v)dv = 1, having dv = 2π sin θdθv 2 dv. When we substitute Equation 2.15 into Equation 2.16, we obtain the following for the second term in Equation 2.16: dg0 (v) cos θ2π sin θdθ v 2 dv nmv NQm (v) g0 (v) + V cos θ dv 4π dg0 (v) = dv. nmv 4 NQm (v) V 3 dv

2

After we apply partial integration in order to replace the derivative of the velocity distribution and take into account the symmetry of the distribution g0 (v), we obtain −

4π 3

m

d 4 v NQm (v) V g0 (v)dv = eE. dv

(2.17)

This is an integral equation that is satisfied for an arbitrary field E and cross sections Qm (v) only if V = − [eE/mNQm (v)v]. In this case we write the expansion for g(v) as g(v) = g0 (v) −

eE cos θ dg0 (v) . mNQm (v)v dv

(2.18)

The second term in Equation 2.18 shows the effect of an external field on the velocity distribution. The cosine term is the lowest-order development of the actual distribution in the velocity space in addition to the symmetric g0 (v) term. Usually the cross section for the momentum transfer of elastic scattering is much greater than the sum of cross sections for other inelastic collisions. So far we have taken into account only elastic collisions. The energy balance of electrons will consist of energy loss in such collisions, as we mentioned before, 2mε/M. Therefore, the total energy loss in elastic collisions in unit time is expressed as 2m 1 2 mv NQm (v)vg(v)dv. M2 The energy gain, on the other hand, is determined by the motion of electrons along the electric field axis: eEv cos θg(v)dv.

© 2006 by Taylor & Francis Group, LLC

14

Plasma Electronics: Applications in Microelectronic Device Fabrication

If we make the balance and apply the expansion 2.18 we obtain m eE cos θ dg0 (v) 2 mv NQm (v)v g0 (v) − dv M mNQm (v)v dv eE cos θ dg0 (v) = eEv cos θ g0 (v) − dv. mNQm (v)v dv

(2.19)

After integration over the polar angles θ and ϕ and by equating the terms inside the integrals we obtain the following first-order differential equation: 4π

m 4π eE dg0 (v) mv 2 NQm (v)vg0 (v) = − eEv . M 3 mNQm (v)v dv

The solution will be obtained by direct integration without making any assumptions on the velocity dependence of the cross section Qm (v): 3m v NQm (v)v 2 g0 (v) = A exp − vdv , (2.20) M 0 eE/m

where A should be determined from the normalization g0 (v)dv = 1. Now we perform the conversion to electron energy distribution f (ε): g0 (v)4πv 2 dv = f (ε)dε;

1 2 mv = ε. 2

(2.21)

The Maxwellian distribution in Equation 2.14 in thermal equilibrium is rewritten as a function of energy ǫ: √ 3ε f M (ε) = A ε exp − , (2.22) 2ε and the mean energy is given by ε = 3kTe /2. In a more general case when there is an electric field but for a constant cross section Qm , we obtain from Equation 2.20 the so-called Druyvestyn distribution: 2 √ ε f D (ε) = A ε exp −0.548 . (2.23) ε

Exercise 2.2.1 Plot the distributions f (ǫ), f M (ǫ), and f D (ǫ) for electrons in Ar at 100 Td. Here the mean energy < ǫ > is 6.47 eV. Figure 2.4 shows the Maxwellian, Druyvestyn, and nonequilibrium ( f 0 ) energy distributions for electrons at the same mean energy. Note the differences among the three distributions. Now we proceed to determine the transport coefficients based on the energy or velocity distributions in velocity space. The drift velocity is defined

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

15

f0

0.10

100 Td 6.47 eV

f (ε)

fD

0.05

fM

0.00

0

5

10

15

20

25

30

Electron energy (eV) FIGURE 2.4 Comparison among f (ǫ), f M (ǫ) , and f D (ǫ) with the same mean energy.

in this chapter as the average velocity of a swarm; that is, vd = vg(v)dv .

(2.24)

v

Combination of expression 2.18 and Equation 2.24 gives eE cos θ dg0 (v) v cos θ g0 (v) − 2πv 2 sin θdθdv vd = mNQm (v)v dv v,θ v2 dg0 (v) 4π eE dv. =− 3 m NQm (v) dv

(2.25)

When we make the transition to the energy distribution in Equation 2.21 we obtain 1 2 ε f (ε) d √ eE dε, (2.26) vd = − 3 m NQm (ε) dε ε where the value of (2/m)1/2 e E of an electron with 1 eV is 5.93 × 107 cm−1 . The transverse diffusion coefficient DT may be derived on the basis of the standard formula λv/3 without field (i.e., for neutral molecules), where λ is the mean free path between two collisions given by (NQm )−1 . Then, by using g(v), DT is expressed in an integral form: 4π v v3 1 DT = g(v)dv = g0 (v)dv. (2.27) 3 NQm (v) 3 NQm (v)

© 2006 by Taylor & Francis Group, LLC

16

Plasma Electronics: Applications in Microelectronic Device Fabrication

The expression by the energy distribution is √ ε 1 2 DT = f (ε)dε. 3 m NQm (ε)

(2.28)

In addition to elastic collisions, there are a number of inelastic processes that may occur. The most significant is the ionization whereby electron collision with a molecule of the background gas produces the pair of a new electron and positive ion. The inelastic processes have a threshold energy as the molecular quantum structure and they are usually characterized by the rate (number of events per second per electron). The ionization rate is calculated from the ionization cross section Qi (ǫ) as NQi (v)vg(v)dv Ri = v eE cos θ dg0 (v) 2π v 2 sin θdθdv = NQi (v)v g0 (v) − mNQm (v)v dv v,θ NQi v(v)g0 (v)4π v 2 dv = v √ 2 NQi (ε) ε f (ε)dε. (2.29) = m ε Another traditional way to describe ionization events is to define the number of ionization events that one electron makes while crossing a unit distance α (also known as Townsend’s ionization coefficient), which can be easily defined as α =

Ri . vd

(2.30)

Normalization of the energy distribution f (ε) used in all calculations of transport coefficients is given by f (ε)dε = 1 (see Equation 2.21). The process that leads to production of a negative ion and loss of the original electron is the electron attachment described by the cross section Qa (ǫ). These two processes, ionization and electron attachment, change the number of electrons and thus are labeled as nonconservative. All these coefficients or rates are directly used in deriving electron transport (fluid) equations. There are also a number of inelastic but conservative (numberconserving) processes (Qk ) that are associated with a wide range of possible energy losses (thresholds). Typically these energy losses εk are quite large, and these processes are much more efficient in controlling the mean energy of electrons than elastic processes, which have an energy loss of approximately 2mε/M. On the other hand, the momentum loss (exchange) is most influenced by the elastic scattering, whose cross section is usually much greater than the sum of the inelastic processes, Qm (ε) ≫ Qi (ε) + Qa (ε) + Qk (ε). Therefore, we measure the contribution of the elastic scattering to the electron transport in terms of the momentum transfer. Rates and spatial excitation

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

17

coefficients are defined in the same way as for ionization. However, these coefficients do not directly enter the transport equations, although they may be used in some corrections and energy balance equations. Thus momentum transfer is dominated by elastic collisions, whereas energy balance is primarily controlled by inelastic processes (with the exception of rare gases and some metallic vapors that have no inelastic processes at low energies). However, it is not practical to use mean energy as the swarm parameter because it cannot be measured directly, so we traditionally define a property of electron swarms known as characteristic energy εk = e DT /µ, where µ = vd /E is the mobility. Characteristic energy has the dimension of energy and is usually within 30% of the mean energy (although sometimes the differences may be up to a factor of two or more). This quantity is usually expressed in units eV or, even more often, by the analogous quantity D/µ, which is expressed in volts. The definition of characteristic energy is given by combining Equations 2.26 and 2.28: √ ε 2 1 f (ε)dε −e E DT DT 3 m NQm (ε) = e E = εk = e ,

f√(ε) µ vd 1 2 d dε e E NQεm (ε) dε 3 m ε which for the thermal Maxwellian distribution in Equation 2.22, in particular, becomes ε − 23 ε dε 2 εk = = ε [eV].

3 ε ε 3 exp − 32 ε dε 2ε NQm (ε)

ε exp NQm (ε)

(2.31)

In the case of the thermal equilibrium where ε = 3kT/2, we obtain the following relation: DT kT = . µ e

(2.32)

This is also known as Einstein’s relation, because Einstein derived an identical equation for the Brownian motion of particles. Moreover, Nernst derived the equation for the electrolytic transport of ions, and Townsend derived it for the electrons in gas discharges even before Einstein, so this relation is often referred to as the Nernst–Townsend relation. This relation is valid only for thermal equilibrium, although it is often used, especially in plasma modeling, for nonequilibrium conditions. In that case, however, it is possible to use corrected forms that are more appropriate. PROBLEM 2.2.2 Discuss the momentum transfer loss of inelastic collision, and compare the influence of the collisional effect between the energy loss and the momentum loss.

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

18

2.2.2 Ion Velocity Distribution and Mean Energy Ions, in principle, follow the same laws that determine the electron transport in gases under the influence of electric or magnetic fields. The basic differences are that the mass of ions Mp is of the same order of magnitude as that of gas molecules M, and that the charge q may not be equal to e and is either positive or negative (here we use the subscript p to denote positive ions, but the conclusions are generally valid for the transport of negative ions as well, only the collisional processes are of a different nature). The energy exchange in elastic collisions is generally significant in an ion swarm. Let us consider an ion swarm with uniform density in real space in a steady state in gases under an external field E = Ek. At low kinetic energy of ions, the collision between the ion and neutral molecule is subject to an induced-dipole interaction, that is, constant mean free time τ , which is given by 1 ǫ0 1/2 1 τ= 1.105 π qN

Mp M Mp + M

1/2

α −1/2 ,

where α is the polarizability of the ion in the dipole interaction with the neutral molecule [2]. The governing equation of the velocity distribution G(v) of ions is then given in the form of the relaxation equation using the BGK approximation [3, 4], dG(v) qE dG(v) G(v) − G 0 (v) = =− . dt Mp dvz τ

(2.33)

Here G 0 (v) is the distribution just after a collision and is approximated by a two-temperature displaced Maxwellian velocity distribution,

G 0 (v) =

β⊥ π

β π

1/2

exp − β⊥ vr2 − β (vz − < vz > + q Eτ/Mp )2 , (2.34)

where β⊥ = 1/(2 < vx2 >), and β = 1/2[< vz2 > − < vz >2 −(q Eτ/Mp )2 ]. Equation 2.33 is then solved as G(v) =

1 vz − < vz > + qEτ/Mp β⊥ exp −β⊥ vr2 − + 2π(qEτ/Mp) qEτ/Mp 4β (qEτ/Mp )2 vz − + qEτ/Mp 1 1/2 qEτ × erfc 2β − , (2.35) Mp 4β (qEτ/Mp )2 2(qEτ/Mp )

where erfc is a complementary error function. The momentum transfer of an ion in the elastic collision is M(v p − V)/ (Mp + M) where V is the velocity of the gas molecule. When we introduce the

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

19

relative velocity vγ = v p − V, we can write the momentum balance as M Mp vγ z k(1 − cos ω)Nσm (vγ , ω)vγ G(vγ )dvγ = qE , (2.36) Mp + M where σm is the differential cross section, ω is the angle of scattering in the center of mass system, and G(vγ ) is the relative velocity distribution of ions. As the collision between the ion and the molecule at low energy is subject to the induced-dipole interaction, (1 − cos ω)Nσm (vγ , ω)vγ d is a constant (Rm ), and Equation 2.36 reduces to Mp M Rm vγ z kG(vγ )dvγ = qE . Mp + M The integral represents the mean velocity, that is, drift velocity vdp in velocity space, which is then equal to vd p =

Mp + M qE . Mp M Rm

(2.37)

When an ion elastically affects a molecule and is isotropically scattered in the center-of-mass system, the averaged energy loss per collision is given by

εl =

Mp M 2 Mp v p − MV 2 . 2 (Mp + M)

T g = 300 K

E / N = 20 Td 1.0

3 200 Td 120 Td

Qm( γ) γ 0.5

2

Collision frequency (a.u.)

Energy distribution f log (log10ε)

(2.38)

3 kT 2 g 0

0.01

0.1

1

10

1

Relative kinetic energy (eV) FIGURE 2.5 Energy distribution at three different E/N and elastic momentum transfer cross section of O− in O2 .

© 2006 by Taylor & Francis Group, LLC

20

Plasma Electronics: Applications in Microelectronic Device Fabrication

PROBLEM 2.2.3 Derive the relation 2.38. In a steady state, the energy balance of the ion swarm in unit time is given by qE · vdp − εl Rm =qEvdp

Mp + M 1 Mp M 2 2 =0, Mp v p −MV 1− (Mp + M)2 Mp M vd2 p (2.39)

which is satisfied when (Mp + M )vd2 p + MV 2 − Mp v 2p = 0 .

By using this relation we obtain the expression of the mean ion energy, which is known as Wannier’s formula [2]: ε p =

1 1 1 Mp v 2p = (Mp + M)vd2 p + MV 2 . 2 2 2

(2.40)

The physical meaning of Equation 2.40 is that the mean energy of ions is separated into three parts: one due to the effect of the electric field given by the drift velocity (first term), one due to the apparent kinetic energy of the encounter molecule (second term), and the thermal energy equal to 3kTg /2 (third term). PROBLEM 2.2.4 When an ion swarm has a shifted Maxwellian velocity distribution in gases with a Maxwell distribution of Tg , derive that the relative velocity distribution between the ion and the molecule G(vr ) also has the Maxwellian velocity distribution.

2.3 Thermal Equilibrium and Its Governing Relations In the case of thermal equilibrium, all processes are balanced by opposite processes, so it is possible to make some quite general distributions both in real and velocity space. In this section, we deal primarily with electrons inasmuch as extensions to other charged particles are trivial. One should bear in mind that in thermal equilibrium all particles should in principle have the same temperature and that the temperature remain constant throughout the whole region, which is in equilibrium. This rule of homogeneous temperature is often difficult to satisfy when it comes to plasmas. We therefore introduce the term “local thermodynamic equilibrium” (LTE), which means that the conditions for thermal equilibrium are satisfied at each point in the plasma but that the value of the effective temperature varies with position. In the

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

21

case of LTE, the temperature is merely a parameter used to apply the laws of thermodynamic equilibrium to a plasma, but this approximation is nevertheless quite effective. 2.3.1 Boltzmann Distribution in Real Space When there is an external electric field, the flux of charged particles consists of drift and diffusion components. Let us consider the z-axis along the field direction. In a weak field, the drift velocity vd is proportional to E and is written as vd = µE using the mobility µ. Therefore, the flux in the z direction is expressed as Γ = nv = nvd − D∇n = nµE − D

∂n k. ∂z

(2.41)

Given that, in thermal equilibrium under randomization, the mean velocity must be zero v = 0, Equation 2.41 leads to a differential equation that connects the spatial density distribution of charged particles n(z) and the spatial potential V(z): µ −µ ∂V(z) 1 ∂n E= = . D D ∂z n ∂z

(2.42)

This equation is solved by direct integration if we define the boundary condition such that for z = 0, the potential is V(0) = 0 and the density is n(0) = n0 . In this case the density distribution is µ n(z) = n0 exp − V(z) . D

(2.43)

We substitute the Einstein relation 2.32 into Equation 2.43 and obtain eV(z) n(z) = n0 exp − , kT

(2.44)

which is known as Boltzmann’s distribution law. This equation describes how the population of charged particles in thermal equilibrium with effective energy kT will be distributed under the space potential V(z). Thus, even for a potential that corresponds to an energy much larger than the thermal energy, there will be a certain number of electrons, although this number is very small. The Boltzmann distribution is often applied to determine the population of excited molecules at different discrete energy levels, in which case it is actually applied to bound electrons and their population at discrete bound levels of atoms or molecules.

© 2006 by Taylor & Francis Group, LLC

22

Plasma Electronics: Applications in Microelectronic Device Fabrication gM

0

Vz

Vx FIGURE 2.6 Equilibrium velocity distribution in vx –v y space.

2.3.2 Maxwell Distribution in Velocity Space The probability that a charged particle will be found in a small volume of velocity space defined by (vx , vx + dvx ), (v y , v y + dv y ), and (vz , vz + dvz ) is given by g(vx , v y , vz )dvx dv y dvz ,

(2.45)

where g(vx , v y , vz ) is the velocity distribution. Under conditions of thermal equilibrium, the distribution g(vx , v y , vz ) must satisfy the following conditions: (a) Each of the components of the velocity distribution must be fully independent of each other, and (b) The velocity distribution must have the same magnitude in all directions, and the mean values of velocities in all directions must be zero. In Figure 2.6 we show a typical distribution function in two dimensions. When we apply conditions (a) and (b), we expand the distribution into three independent functions g(vx , v y , vz )dvx dv y dvz = G(vx )dvx G(v y )dv y G(vz )dvz = g(v 2 )dvx dv y dvz ,

(2.46)

where the magnitude of the velocity v is defined by v 2 = vx2 + v 2y + vz2 , and the distribution in terms of v 2 is then defined as g(v 2 ) = G(vx )G(v y )G(vz ). We must choose the function g to satisfy the functional Equation 2.47.

© 2006 by Taylor & Francis Group, LLC

(2.47)

Phenomenological Description of the Charged Particle Transport

23

Exercise 2.3.1 Show that the solution to Equation 2.47 is g vx2 = a 2exp −αvx2 ,

(α > 0).

(2.48)

Let us observe a special case of one-dimensional motion where v y = vz = 0. In this case, Equation 2.48 is written as g vx2 = G(vx )G(0)G(0) = a 2 G(vx ),

where a = G(0) and, by the same argument, g(v 2y ) = a 2 G(v y ) and g(vz2 ) = a 2 G(vz ). Therefore, Equation 2.47 is rewritten as 1 g(v 2 ) = g vx2 + v 2y + vz2 = 6 g vx2 g v 2y g vz2 . a

(2.49)

Note that the functional Equation 2.49 satisfies Equation 2.48 as the solution of each component. Therefore, we obtain g(v 2 ) = g vx2 + v 2y + vz2 = A exp −α vx2 + v 2y + vz2 ,

(2.50)

where the value of the constants A and α must be determined. When we normalize the velocity distribution g(vx , v y , vz ) to 1, 1 = g vx2 + v 2y + vz2 dvx dv y dvz ∞ ∞ ∞ exp − αvz2 dvz exp − αv 2y dv y exp − αvx2 dvx = A −∞

3 π , = A α

−∞

−∞

and therefore 3/2 α A= . π Now we apply the distribution g(vx , v y , vz ) to determine the mean kinetic energy of charged particles with mass m at temperature T according to the conditions of thermodynamic equilibrium α 3/2 m 3 1 2 2 2 kT = mv 2 = vx2 + v 2y + vz2 e −α(vx +v y +vz ) dvx dv y dvz 2 2 π 2 α 3/2 3m 1 π 3/2 = , π 2 2α α

© 2006 by Taylor & Francis Group, LLC

24

Plasma Electronics: Applications in Microelectronic Device Fabrication

which leads directly to α=

m 2kT

and

A=

m 3/2 . 2π kT

(2.51)

We use the constants 2.51 to obtain the final form of the Maxwellian distribution from Equation 2.50, m 3/2 m vx2 + v 2y + vz2 . (2.52) exp − g M = g(vx , v y , vz ) = 2π kT 2kT In the previous derivation of Maxwell’s velocity distribution, we used the

∞

∞ √ √ 2 2 following integrals: 0 e −αx d x = 1/2 π/α, and 0 x 2 e −αx d x = 1/2α π/α. In place of Maxwell’s velocity distribution, the Maxwellian speed distribution F (v) is defined as g M (v 2 )v 2 sin θdθdϕdv F (v)dv = θ

ϕ 2

= 4πv g M (v 2 )dv .

(2.53)

The following averaged velocities are defined for the Maxwellian speed distribution: (i) most probable speed v˜ , (ii) root mean square velocity v 2 , and (iii) mean speed v. These are obtained as follows. i. The most probable speed v˜ is obtained from the condition dF /dv = 0, and its value is 2kT 1/2 v˜ = . (2.54) m ii. The root mean square velocity (v 2 )1/2 is the square root of the mean value of v 2 : 3kT v 2 F (v)dv = v 2 = . (2.55) m iii. The mean speed is obtained as v =

vF (v)dv =

8kT . πm

(2.56)

We can easily compare the magnitudes of these three mean values as v˜ < v < v 2 . (2.57)

Figure 2.7 shows the plot of the normalized Maxwell speed distribution F (v) and gives the values of the three possible choices for averaging the velocity. Different physical quantities may be averaged with different powers of

© 2006 by Taylor & Francis Group, LLC

Phenomenological Description of the Charged Particle Transport

25

0.8

F(v)

0.6

0.4

0.2

0

0

0.5

1.0

1.5

2.0

v ~ v

兹 ~ ~ v v Normalized velocity FIGURE 2.7 Normalized Maxwell speed distribution and the relationships among these averaged velocities.

speed and will be associated with different effective velocities. It is clear that the mean speed may be used to establish the momentum transfer, that the mean square speed is relevant for consideration of the mean energy, and that the most probable speed is the peak of the probability distribution. Although one may directly produce averages of physical phenomena, associating them with mean values of speed provides a better physical insight into the phenomenon under consideration.

References 1. 2. 3. 4.

Parker, J.H., Jr. and Lowke, J.J. 1969. Phys. Rev. 181:290–301. Wannier, G.H. 1953. Bell Syst. Tech. J. 32:170. Bhatnagar, P.L., Gross, E.P., and Krook, M. 1954. Phys. Rev. 94:511. Whealton, J.H. and Woo, S.B. 1971. Phys. Rev. A 6:2319.

© 2006 by Taylor & Francis Group, LLC

3 Macroscopic Plasma Characteristics

3.1 Introduction Plasma is a compound phase with electric quasi-neutrality, generally consisting of electrons, positive ions, and neutral molecules. In particular, strongly ionized plasma consists of electrons and positive ions. The phase is called collisionless plasma, as the electron (ion) has few short-range binary collisions with neutral molecules, and the plasma system is subject to long-range Coulomb interactions. When the short-range two-body collision is major, the system is collisional, that is, collision dominated. Collisionless plasma shows unique characteristics. We briefly describe the characteristics [1–3].

3.2

Quasi-Neutrality

The presence of electrons and ions with density ne and np will produce an electric field subject to Poisson’s equation, divE = e

n p − ne , ǫ0

(3.1)

where e(>0) is the elemental charge and ǫ0 is the permittivity (dielectric constant) of vacuum. A plasma where the density of positive ions is exactly equal to that of electrons has no electric field. Naturally, based on the great difference in mass between the electron and positive ion, a quasi-neutral state with np ∼ ne is maintained in plasmas on a macroscopic scale. This causes a finite positive plasma potential with respect to the wall earthed to the ground. Notice that a low-temperature plasma externally excited by direct current (DC) or rf power source has a plasma production region and holds a structure bifurcating into a positive ion sheath with a high field to produce an electron– ion pair and a bulk plasma with quasi-neutrality.

27 © 2006 by Taylor & Francis Group, LLC

28

Plasma Electronics: Applications in Microelectronic Device Fabrication

∆

plasma (ne~np) p-ion electron

d

FIGURE 3.1 Plasma fluctuation and the charge separation in a collisionless plasma with macroscopic quasineutrality.

3.3 Charge-Separation in Plasmas 3.3.1 Spatial Scale of Charge-Separation In a bulk plasma, quasi-neutrality will be macroscopically kept in space and time. The quasi-neutral condition, however, will be locally disturbed by the intrinsic random motion of the charged particles in a plasma. We first assume the presence of a localized space–charge layer as shown in Figure 3.1 as a result of the random fluctuation of charged particles in a bulk plasma with macroscopic quasi-neutrality. In one-dimensional space, the local field and the potential difference caused by the two layers is given by Poisson’s Equation 3.1 as E=

en ε0

and

V=

end , ε0

where n(∼ ne ∼ n p ), is the thickness of each of the layers, and d is the distance between layers. In a collisionless plasma, when the potential energy of the charged particle in the layers is less than the random thermal energy, kTe or kTp , e

end < kTe , ε0

(3.2)

the local disturbance of the charge-separation will be kept. Considering the scale of ∼ d, ε0 kTe 1/2 ≡ λ D. (3.3) d< ne 2 Here, λ D defined in Equation 3.3 gives a microscopic maximum spatial scale for the charge-separation and is known as the electron Debye (shielding) length or Debye radius.

© 2006 by Taylor & Francis Group, LLC

Macroscopic Plasma Characteristics

29

Exercise 3.3.1 Calculate the Debye length when plasma density and electron temperature in a collisionless plasma are 1016 m−3 and 3 eV, respectively. kT[eV] 1/2 ε0 kTe 1/2 3 ∼ 7.43 × 10 × [m]. (3.4) λD = ne 2 n[m−3 ] That is, 3

λ D = 7.43 × 10 ×

3 1016

1/2

= 1.29 × 10−4 [m].

3.3.2 Time Scale for Charge-Separation As there exists a huge difference in mass between the electron and the positive ion, the electron layer in Figure 3.1 will go toward the positive ion layer under the Coulomb force. In a collisionless plasma, the local drift motion of the electron layer passes through the massive positive layer by inertia, and the pair of charged layers is oppositely reformed. The electron layer will continue to vibrate around the massive positive ions in the absence of two-body collision between the neutral molecules. The temporal motion is described by the harmonic oscillation, m

e 2n d2 (t) = − (t). dt 2 ε0

(3.5)

The oscillation is known as the electron plasma oscillation or Langmuir oscillation. The frequency of the electron plasma oscillation, electron plasma frequency, is numerically given as ωe =

e 2n mε0

1/2

1/2 −1 ∼ 56.4 n[m−3 ] [s ],

(3.6)

The time scale of the charge-separation in a collisionless plasma τes , given by ωe−1 , has the relation with λ D , λD kTe 1/2 = ≈ Ve , (3.7) τes me where Ve is the electron thermal speed. PROBLEM 3.3.1 Discuss that the plasma frequency ω p of the ion with charge Ze is defined in the same way as the electron, ωp =

© 2006 by Taylor & Francis Group, LLC

Z2 e 2 n p Mp ε0

1/2

=Z

m Mp

1/2

ωe .

(3.8)

30

Plasma Electronics: Applications in Microelectronic Device Fabrication

Also discuss that practical plasma oscillation is given by (ωe2 + ω2p )1/2 = ωe (1 + z2 me /Mp )1/2 . PROBLEM 3.3.2 Estimate the values of ωe and ω p in collisionless Ar plasma with density 1016 m−3 . PROBLEM 3.3.3 A plasma is externally irradiated by an electromagnetic wave with frequency of ω. Derive the condition that the external disturbance does not immerse deeply in the plasma.

3.4 Plasma Shielding 3.4.1 Debye Shielding When a material is inserted into a plasma, the surface of the material is immediately charged regardless of metal or insulator, and it is electrically shielded from the surrounding plasma (see Figure 3.2). We consider the spatial scale of the shielding caused by a spherical metal inserted into a plasma with zero space potential. The potential distribution close to the metal is obtained by Poisson’s equation as a function of radial distance r , ∇ 2 V (r ) =

1 d r 2 dr

dV (r ) e n p − ne . r2 =− dr ε0

(3.9)

There are number of electrons and ions in the shielded region in front of the sphere. The number density of electron and ions, ne (r) and n p (r), follows the V n np(r) n0 ne(r) -V(r) 0

FIGURE 3.2 Plasma shielding of a charged sphere.

© 2006 by Taylor & Francis Group, LLC

λD

q 4πε0r r

Macroscopic Plasma Characteristics

31

Boltzmann distribution in space, discussed in Section 2.3, eV (r ) , n p = n0 exp − kTp ne = n0 exp

eV (r ) kTe

,

(3.10)

(3.11)

where n0 is the quasi-neutral plasma density far from the influence of the impurity metal. At a long distance from the metallic surface satisfying eV(r ) ≪ kTe , kTp , the exponential terms in Equations 3.10 and 3.11 will be expanded by the Taylor series, and we obtain 1 d r 2 dr

eV (r ) dV (r ) en0 eV (r ) r2 =− exp − − exp dr ε0 kTp kTe ≈

e 2 n0 Tp + Te V (r ) . ε0 kTp Te

(3.12)

The general converging solution of Equation 3.12 is r A exp − , V (r ) = r λD where A (> 0) is constant and λ D is λ2D =

ε0 kTp Te . Tp + Te

e 2 n0

(3.13)

The potential close to the metal with a surface charge q 0 is expressed by the Coulomb potential, q 0 /4πǫ0r . Then, the final form is r q0 exp − . (3.14) V (r ) = 4π ε0r λD Noted that V(r ) is negative due to q 0 (< 0) under me ≪ Mp . Equation 3.14 shows that the scale length of the shielding is on the order of the Debye length λ D . 3.4.2 Metal Probe in a Plasma A small metallic probe with variable potential is inserted into a plasma in a steady state. A static shielding sheath is formed there without plasma production or loss. We consider the voltage-current characteristics I (V) when the potential at the probe is externally varied with respect to the wall of the plasma. They are known as probe characteristics in a plasma. Figure 3.3 shows typical probe characteristics.

© 2006 by Taylor & Francis Group, LLC

32

Plasma Electronics: Applications in Microelectronic Device Fabrication I insulator

metal

a

b plasma

(I)

(II)

λD

VS

Vfl d (III)

VS c

V

Isi

FIGURE 3.3 Ideal Langmuir probe characteristics in a plasma.

When the probe has a sufficiently large surface, a saturated electron current will be collected at V > Vs where Vs is the space potential of the plasma. Practically due to the finite surface area, I (V) characteristics at V > Vs will increase gradually as shown in Figure 3.3. The probe potential at c where the current is zero is named the floating potential. The name comes from the zero net flux at the surface of an insulator inserted into a plasma. The probe will be given an absorbing boundary that absorbs electrons and ions incident on the surface. At the region bc that satisfies V < Vs , the electron sheath will be formed in front of the probe (Vs − V < 0). The electron incident right on the probe surface will be retarded by the field, and only the electron with kinetic energy mv 2 /2 > |e(Vs − V)| will be absorbed as the electron current. We consider that the electron has the Maxwellian velocity distribution with temperature Te in the plasma and has no collision in the passive sheath in front of the probe. Then the electron probe current will be given by 3/2 ∞ ∞ ∞ m I (V) = ene S 2e|V | vz p 2π kTe vx =−∞ v y =−∞ vz = m

m(vx2 + v 2y + vz2 ) × exp − dvx dv y dvz 2kTe 3/2 ∞ ∞ mv 2y m mvx2 = ene S dvx exp − exp − 2π kTe 2kTe 2kTe −∞ −∞ ∞ mvz2 × dv y dvz exp − 2kTe vz √ 3/2 √ eVp m 2 π 2 π 1

exp − = ene S , (3.15) m m m 2π kTe kTe 2 2kT 2 kT e 2kTe e

© 2006 by Taylor & Francis Group, LLC

Macroscopic Plasma Characteristics

33

where S is the effective area of the probe. By considering the mean speed ve = (8kTe /π m)1/2 in Equation 3.15, the probe I (V) characteristics are described by V − Vs ene ve . S exp I (V) = 4 kTe

(3.16)

Equation 3.16 shows that the electron temperature Te in the plasma is obtained by the gradient of ln I (V ). At ab satisfying V > Vs (electron saturated region), all electrons approaching the probe will be collected, the lower limit of the integral in Equation 3.15 will be zero, and I =

ene ve S. 4

(3.17)

Equation 3.17 means that electron current consists of ne /2 in onedimensional position space and ve /2 in velocity space. In principle, by using the electron temperature Te , the plasma density (electron density) will be given by the saturated curve, Equation 3.17, in the region of bc. The probe having one small metal is called the single probe or the Langmuir probe. Other types of probes are also used as a simple tool for plasma diagnostics.

Exercise 3.4.1 Discuss the pressure condition that the electron temperature Te is estimated from the curve in region (II) in bc in Figure 3.3. As a typical plasma we assume, ne = 1015 m−3 , kTe = 3.0 eV. Then, the Debye length is 3

λe = 7.43 × 10 ×

kTe [eV] ne [m−3 ]

1/2

√ 7.43 × 103 × 3 √ = 4.07 × 10−4 [m]. = 10 × 107

The mean speed of electrons is ve = (8kTe /πm)1/2 = 6.71 × 107 ×

kTe [eV] ≈ 1.16 × 106 [ms−1 ].

The collision rate R is roughly approximated at 107 p[Pa s−1 ], and the flight time of the electron in the static sheath in front of the probe is 4.07 × 10−4 1 λe ≈ ≈ 3.51 × 10−10 ≪ ve 1.16 × 106 R

≈

1 107 p

Therefore, the collisionless condition is obtained as p ≪

1 1 × ≈ 2.85 × 102 [Pa]. 107 3.51 × 10−10

© 2006 by Taylor & Francis Group, LLC

.

34

Plasma Electronics: Applications in Microelectronic Device Fabrication

wall

plasma (ne ~ np)

A+ e

Γe ~ Γ p Ea FIGURE 3.4 Fluxes of charged particles in front of a wall.

3.5 Particle Diffusion 3.5.1 Ambipolar Diffusion We consider a plasma without production in a vessel far from a plasma source. Electrons and ions in the plasma have a random motion with a kinetic energy. Even in the same kinetic energy (temperature) between electrons and ions, the electron flux incident on the wall as a result of the random motion will be much higher than the massive positive ion, and the relation Ŵe ≫ Ŵ p is immediately attained (see Figure 3.4). Then, an electric field that accelerates positive ions and decelerates electrons toward the wall surface (i.e., ambipolar field) will be formed, and in a macroscopically steady state, the charged particles’ flow to the wall with the relation (3.18)

Ŵe ≈ Ŵ p

is realized. We deal semiquantitatively with the ambipolar diffusion phenomena caused between the static plasma and the surrounding reactor wall. Each of the fluxes of electrons and ions with density, ne and n p , is given, Γe = ne ve = ne vde − De

dne , dr

dn p Γ p = n p v p = n p vd p − Dp , dr

(3.19)

(3.20)

where v j , vd j , and D j are the mean velocity, drift velocity, and diffusion coefficient of electrons and positive ions, respectively. In a steady state as mentioned above, the electron and ion have the same velocity toward the wall, ambipolar diffusion velocity va , va = −Da

© 2006 by Taylor & Francis Group, LLC

1 dn , n dr

(3.21)

Macroscopic Plasma Characteristics

35

where Da is the ambipolar diffusion coefficient given by Da =

µ p De + µe Dp , µ p + µe

(3.22)

where µe and µ p are the mobility of electrons and ions, respectively. In particular, at low reduced field the mobility may be constant. In thermal equilibrium of electrons and ions with Maxwellian velocity distribution, the ambipolar diffusion coefficient is expressed in terms of the electron and ion temperatures, Te and Tp , Dp 1 + TTep . Da = (3.23) µ 1 + µep In the case of Te ≫ Tp ,

Da = Dp

Te . Tp

(3.24)

In particular, in Te = Tp , Da = 2Dp .

(3.25)

PROBLEM 3.5.1 Derive the ambipolar diffusion coefficient, Equation 3.22, and show that the ambipolar diffusion field is written as Ea =

Dp − De 1 dn . µe + µ p n dr

(3.26)

3.5.2 Spatial and Time Scale of Diffusion We consider the time and spatial scale of particles in a collisionless or collisional plasma. In the case where the production and loss of the particle is negligible, the continuity equation of the particles is ∂ n(r, t) = D∇ 2 n(r, t). ∂t

(3.27)

The diffusion Equation 3.27 implies that the spatial distribution and time scale of the diffusion are determined by the geometry of the reactor. Here, we divide n(r, t) into two independent functions n(r) and T(t). Then we have 1 1 1 dT (t) = D∇ 2 n(r) ≡ − . T (t) dt n(r) τD Here, τ D (> 0) is the separation constant.

© 2006 by Taylor & Francis Group, LLC

(3.28)

36

Plasma Electronics: Applications in Microelectronic Device Fabrication

PROBLEM 3.5.2 Discuss the physical meaning that the r.h.s. of Equation 3.28 has negative value. Equation 3.28 is written as T (t) = T0 exp(−t/τ D ) ,

(3.29)

and ∇ 2 n(r) = −

1 n(r) n(r) = − 2 , Dτ D

where 2 = Dτ D .

(3.30)

Equation 3.30 shows that the spatial characteristics of particle diffusion 2 are determined by the geometrical boundary condition. is called the characteristic diffusion length and τ D the diffusion decay time. PROBLEM 3.5.3 Calculate the characteristic diffusion length in the case where the reactor geometry is (a) an infinite parallel plate (separation of d0 ), (b) an infinite rectangle (two sides, d0 and l0 ), (c) a cylinder (radius r0 ), and (d) a sphere (radius r0 ), respectively. 1 = (π/d0 )2 ; infinite plates, 2 = (π/d0 )2 + (π/l0 )2 ; infinite rectangle, = (2.405/r0 )2 ; cylindrical, = (π/r0 )2 ; spherical.

(3.31)

PROBLEM 3.5.4 Neutral molecules diffuse in three-dimensional space without boundary. Calculate the number density of molecules n(r, t) in Equation 3.27. In particular, when the initial condition of the density is given by n(r, t = 0) = δ(r), derive the density distribution n (r, t) =

1 4π Dt

3/2

r2 exp − . 4Dt

(3.32)

The above stochastic process is known as the Wiener process and the density distribution is the normal distribution N(0, 2Dt) with mean value of 0 and variance of 2Dt. The characteristic diffusion length D is much influenced by the loss mechanism of the particle. For example, we consider the electron diffusion in electronegative gases with density N and with the electron attachment rate coefficient ka . Then the electron continuity equation is ∂ n(r, t) = −ka n(r, t)N + D∇ 2 n(r, t). ∂t

© 2006 by Taylor & Francis Group, LLC

(3.33)

Macroscopic Plasma Characteristics

37

Equation 3.33 is reduced to 1 dT (t) 1 1 = −ka N + D∇ 2 n(r) ≡ − . T (t) dt n(r) τ Dr

(3.34)

As a result, the characteristic diffusion length changes from Equation (3.30) to ka N −1 1 2 − . (3.35) = Dτ D D

Exercise 3.5.1 Molecules excited to an optically forbidden level are called metastables and have a long lifetime. The metastable state is de-excited without photoemission when it reacts with the wall with absorbed molecules and exhausts its excess inner energy to the wall. Estimate the effective lifetime of metastables in a reactor of volume Vol with reflecting wall. Reflection coefficient γr at the wall is given by γr = 1 − St , where St is the surface sticking coefficient, 2 2 2 2Vol (2 − β) τ= + . (3.36) π D St vβ

3.6 Bohm Sheath Criterion 3.6.1 Bohm Velocity A fluid description of a region between a plasma and material boundary (i.e., plasma sheath) will give a simple physical image from the ambipolar diffusion transport of the charged particle. As described in the previous section, the net current to the surface disappears as a result of the ambipolar diffusion in the steady state, and the surface is kept at the floating potential. The electron and ion dynamics are valid in the sheath under the following assumptions. 1. The frequency of the external power source is much higher than the collision rate ω ≫ R (i.e., the collisionless sheath).

2. In the momentum continuity equation (see details in Chapter 5), the pressure gradient dP(z)/dz = kTdne (z)/dz is dominant for electrons, whereas other terms except the pressue are predominant for ions. 3. The ion flux is constant, that is, n p (z)vdp (z) = const, without ionization and recombination in the collisionless sheath. Under these conditions, the momentum equations of electrons and ions in the steady state are, respectively, me

© 2006 by Taylor & Francis Group, LLC

dP(z) dV(z) = −ene (z) , dz dz

(3.37)

38

Plasma Electronics: Applications in Microelectronic Device Fabrication

Mp n p (z)vd p

dV(z) dvdp = en p (z) . dz dz

(3.38)

From the above ion flux continuity and Equation 3.38, we obtain n p (z) =

n p (z0 )vd p (z0 ) vdp (z)

= n p (z0 )

1 1−

2e[V(z)−V(z0 )] Mp vd p (z0 )2

1/2 .

In a local thermal equilibrium (LTE) under Equation 3.37, the electron number density has the Boltzmann distribution (see Section 2.3). The region from a bulk plasma/sheath boundary at z0 to the sheath terminal, consisting of electrons and ions, is expressed by Poisson’s equation dV(z)2 e = − [n p (z) − ne (z)] dz2 ε0

ene (z0 ) e[V(z0 ) − V(z)] 2e[V(z) − V(z0 )] −1/2 =− − exp − 1− . ε Mp vd p (z0 )2 kTe In the sheath, positive ions are dominant; that is, n p (z) − ne (z) > 0. Accordingly, 2e[V(z) − V(z0 )] −1/2 e[V(z) − V(z0 )] 1− − exp > 0. Mp vdp (z0 )2 kTe

(3.39)

In particular, at the position z close to the boundary between the bulk plasma and the sheath, e[V(z) − V(z0 )] ≪ kTe will be satisfied. By using a Taylor expansion of the exp term in Equation 3.39, 2e[V(z) − V(z0 )] 2e[V(z) − V(z0 )] 1− . < 1− kTe Mp vd p (z0 )2 Finally, we have the initial directional velocity of ions just entering the sheath from the bulk plasma (i.e., Bohm velocity): vd p (z0 ) >

kTe Mp

1/2

(3.40)

.

3.6.2 Floating Potential ′

Here, we define the sheath edge at the position z0 where vd p is equal to (kTe /Mp )1/2 . ′

Vplasma − V(z0 ) =

© 2006 by Taylor & Francis Group, LLC

′

Mp vd p (z0 )2 kTe = . 2e 2e

(3.41)

Macroscopic Plasma Characteristics

39 ′

Therefore, the electron density at z0 is e[Vplasma − V(z0 ′ )] ′ = 0.61ne0 . ne (z0 ) = ne0 exp − kTe

(3.42)

When a small metal plate isolated electrically from the reactor is immersed in a bulk plasma with a plasma potential Vplasma and electron temperature Te , the surface potential Vf l , called the floating potential, is given at the condition of zero net current eŴe = eŴ p as kTe Mp Vf l = Vplasma − . (3.43) ln 2e 2.3m

Exercise 3.6.1 Derive the above expression. Although eŴe = eŴ p is satisfied, positive ions have a strong directional flux, and electrons show an isotropic flux with random speed < ve >= (8kTe /π m)1/2 (see Chapter 2). Therefore, e[Vplasma − Vf l ] 1 kTe 1/2 ′ ne0 exp − . < ve >(= (n p (surf )vd p (surf )) = n p (z0 ) 4 kTe Mp ′

′

By using Equation 3.42 under ne (z0 ) ∼ n p (z0 ), e[Vplasma − Vf l ] π m 1/2 − = ln 2.4 . kTe 8Mp Finally we will obtain Equation 3.43.

References 1. Chen, F.F. 1984. Introduction to Plasma Physics and Controlled Fusion, Vol. 1. New York: Plenum. 2. Golant, V.E., Zhilinsky, A.P., and Sakharov, I.E. 1980. Fundamentals of Plasma Physics. New York: John Wiley & Sons. 3. Cherrington, B.E. 1979. Gaseous Electronics and Gas Lasers. Oxford: Pergamon.

© 2006 by Taylor & Francis Group, LLC

4 Elementary Processes in Gas Phase and on Surfaces

A large number of individual elementary processes are important in determining the kinetics of low-temperature plasma. The basic processes in the gas phase include collisions of electrons, ions, and fast neutrals with molecules of the feed gas and with surfaces. Collisions between charged particles, collisions between excited particles, and collisions of radicals with gas molecules are also important. In addition, one needs to consider ion–molecule reactions and different types of photon interactions with molecules, excited species, and surfaces. It is essential to understand the nature of elementary processes in order to investigate the collective phenomena and kinetics of plasmas as a whole. Electron collisions are particularly important, because they can produce new electrons and ions under most conditions and also produce the chemically active species required for numerous plasma applications. Gas phase collisions are dominant in low-temperature plasmas. If we change the background feed gas in a low-temperature plasma reactor, the nature of the plasma will change considerably. In highly ionized plasmas, the nature of the gas is relatively unimportant (unless thresholds for ionization change dramatically), because they are dominated by charged particle collisions. In an earlier chapter we have seen that the motion of charged particles between two collisions is classical unless the gas density is very high. However, quantum effects do occur for a very brief period of time during the collision. Thus we must briefly review the quantum representation of particles and particle beams and quantum scattering theory [1.2]. Classical collision theory is also reviewed in this chapter, because it proves useful in some cases. We then identify the basic characteristics of different collision processes [3.4.5]. We analyze electron collisions over a very wide range of energies while briefly discussing ion and fast neutral scattering over ranges relevant for practical plasmas. Finally, we discuss collisions and reactions of neutrals and ions at energies close to room temperature, which is appropriate for low-temperature plasmas. Numerous elementary processes take place at surfaces. In this chapter we consider surfaces as a source of charged particles both through collisions and

41 © 2006 by Taylor & Francis Group, LLC

42

Plasma Electronics: Applications in Microelectronic Device Fabrication

arising from surface heating and surface reactions. A special group of surface processes associated with plasma etching are also described briefly here and in greater detail in Chapters 8 and 12.

4.1 Particles and Waves 4.1.1 Particle Representation in Classical and Quantum Mechanics An individual particle in classical physics is represented by its mass m, its energy ε, and its momentum p. Assuming that the particle is moving along √ the z-axis, the components of the momentum are p( px = 0, p y = 0, pz = 2mε). In Figure 4.1 we show a classical representation of a moving particle. In quantum mechanics, the motion of a particle is described by a matter wave (de Broglie wave). Individual atomic and elementary particles are subject to quantum mechanical laws; therefore, their properties are subject to the Heisenberg uncertainty principle, which relates the uncertainty of the spatial coordinate r with the uncertainty of the momentum p. The uncertainty relation can be written as r · p ≥

h (= h¯ ), 2π

(4.1)

where h is Planck’s constant. In quantum mechanics, the energy and the momentum of the particle (ε, p) are related to the angular frequency and the wave number of the plane wave (ω, k) through ε = h¯ ω

p = h¯ k.

(4.2) (4.3)

A particle moving along the z-axis is represented as a plane wave (r, t) = A0 exp[ j (k · r − ωt)],

(4.4)

as shown in Figure 4.1. Here, A0 is the amplitude, and k · r = const expresses the wave front. This plane wave is fully defined by two pairs of quantities, (ω, k). PROBLEM 4.1.1 Show that the plane wave satisfies the uncertainty principle.

Exercise 4.1.1 Calculate the de Broglie wavelength λe for the electron that has crossed a potential drop V0 (see Figure 4.2).

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces wave front

43

k . r = const.

at t = t0

r p z 0 k

m (particle)

λ

λ

(plane wave)

FIGURE 4.1 Classical and quantum (plane wave) description of a particle with the mass m and momentum p(0, 0, pz ).

The energy of a particle is equal to the potential drop V0 times the elementary charge e, so we can obtain the momentum of the particle as a function of V0 : ε = p 2 /2m = eV0 . Bearing in mind the de Broglie relation in Equation 4.3, we obtain p = h¯ k =

h 2π h · = . 2π λe λe

Finally, we have λe = =

h (2me V0 )1/2 6.63 × 10−34 = (2 × 9.11 × 10−31 × 1.6 × 10−19 × V0 )1/2 C

1.50 V0 [V]

1/2

[nm].

A G λe

electron beam

emitter

V0 FIGURE 4.2 A stationary electron beam with mono-energetic energy eV0 produced in a vacuum.

© 2006 by Taylor & Francis Group, LLC

(4.5)

44

Plasma Electronics: Applications in Microelectronic Device Fabrication

Exercise 4.1.2 Obtain the dispersion relation between the frequency and the wave number for a plane matter wave (r, t) in free space (see Equation 4.4). If we start from the time- and space-dependent Schrodinger ¨ equation H (r, t) = j¯h

∂ (r, t), ∂t

(4.6)

with the Hamiltonian for a free particle given by H=−

h¯ 2 2 ∇ + V(r, t), 2m r

(4.7)

then the solution in free space (V = 0) to this system with the plane wave in Equation 4.4 gives us h¯ 2 k 2 (r, t) = h¯ ω (r, t). 2m Finally, we obtain the dispersion relation (between the wave number k and the angular frequency ω) ω(k) =

h¯ k 2 . 2m

(4.8)

PROBLEM 4.1.2 The wave-front of a plane wave is defined as k · r = const at time t, and in that case the wave front is perpendicular to the wave vector k. Show that a spherical wave with a circular wave front is expressed as (r, t) =

A0 exp[ j (k · r − ωt)]. r

(4.9)

4.1.2 Locally Isolated Particle Group and Wave Packets An isolated group of particles moving in the z-direction is expressed as a wave packet. To obtain a wave packet, we can sum a group of plane waves in respect to k within the boundaries (k0 ± k). It is assumed that (k − k0 ) = ξ ≪ k0 . The summation may be converted to an integral as follows: (z, t) = =

© 2006 by Taylor & Francis Group, LLC

k 0 +k

(z, t; k)

k0 −k

k0 +k

k0 −k

A(k) exp[ j (kz − ωt)]dk.

(4.10)

Elementary Processes in Gas Phase and on Surfaces

45

λ0

z

FIGURE 4.3 Wave packet of a group of particles moving in the z-direction.

Furthermore, we apply the Taylor expansion to the dispersion relation around k = k0 : ω = ω(k − k0 ) dω 1 d 2 ω = ω0 + (k − k0 )2 + · · ·, (k − k ) + 0 dk 0 2 dk2 0

and, as a result, we obtain k dω A(k) exp j (k0 + ξ )z − j ω0 + ξ t dξ (z, t) = dk 0 −k k dω ξ t dξ. exp j ξ z − ∼ A(k0 ) exp[ j (k0 z − ω0 t)] dk −k

0

By combining the above equation with k e ja k − e − ja k 2 sin(a k) exp( ja θ )dθ = = ja a −k

we obtain sin (z, t) ∼ 2A(k0 )

t k dk 0 exp[ j (k0 z − ω0 t)]. z − dω t dk

z−

dω

(4.11)

0

In Figure 4.3, we show a plot of Equation 4.11. The locally isolated particle group is represented by a strongly modulated wave with the fundamental frequency ω0 (k0 ). There is an envelope with a maximum in the position where a particle should be in a classical model (i.e., the center-of-mass) and the undulations extend approximately over the one wavelength λ0 . The exponential term in Equation 4.11 determines the phase of the wave.

© 2006 by Taylor & Francis Group, LLC

46

Plasma Electronics: Applications in Microelectronic Device Fabrication

The position of the maximum of the wave packet (z = dω/dk|0 t) moves with a velocity dz dω , (4.12) = Vg = dt dk 0

also known as the group velocity. For the particular example given in Equation 4.8, the result is Vg =

d dk

h¯ k 2 h¯ k0 = . 2m 0 m

(4.13)

If we remember that p(= mv) = h¯ k0 , then Vg corresponds to the classical velocity v.

Exercise 4.1.3 Derive the phase velocity of the wave packet in Equation 4.11. The phase in Equation 4.11 is = k0 z − ω0 t = const, and the phase velocity Vp is equal to

Vp =

d dz = dt dt

ω0 t k0

=

ω0 . k0

Using Equation 4.8 allows the phase velocity to be related to the group velocity Vp =

1 h¯ k02 h¯ k0 Vg = = . k0 2m 2m 2

PROBLEM 4.1.3 A pulsed electron beam is formed in a vacuum by using the electrical shutter between grids A and G in Figure 4.2. The half width of the group z is 1 mm, and V0 is 150 V. Discuss the uncertainty principle using vz /vz .

4.2 Collisions and Cross Sections An atomic collision is a stochastic process that takes place when two particles approach one another at a sufficiently short distance so that their interaction through different forces becomes appreciable. In weakly ionized plasmas, we deal mainly with two-body, short-range collisions, which are then regarded as occurring very quickly and over a very short distance as compared with the mean (free) time and mean (free) path between successive collisions. Because

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces scattered beam

(a)

dΩ

scattered spherical wave

dΩ

incident plane wave

v m incident particle

(b)

47

z

z

M target molecule

FIGURE 4.4 Classical (a) and quantum (b) representation of a collision.

the Coulomb force has infinite range, collisions between charged particles are described as many-body, long-range collisions. There is also a whole group of three-body processes where the third particle is required to satisfy the conservation laws. A three-body process occurs, for example, when one of the species is left in an excited state after the two-body collision and the excited species needs a subsequent collision with a third body to transfer its extra energy for stabilization. Three-body processes will occur only if the gas density is sufficiently high. Here we deal primarily with two-body, short-range collisions. In classical mechanics, particles have definite identities, positions, and velocities (momenta). The classical collision of a light projectile of mass m and velocity v on a heavy target of mass M is depicted in Figure 4.4a. After the collision the light particle will be scattered at an angle θ with respect to its original direction z and into the solid angle d(= sin θdθ dφ). It will have a definitive velocity and path, and the differential cross section σ (θ, φ, ε)d will give the probability of scattering into the solid angle d. In the quantum case, an incoming free particle will be represented as a plane wave described by Figure 4.4b and Equation 4.4. Then, as a result of spherical scattering, the outgoing wave will be given by the sum of the plane and the spherical waves. 4.2.1 Conservation Laws in Collisions First we assume that the collision event occurs in a force-free space, in which the momentum and energy of the two particles are conserved. This is true, for example, for an electron collision with a neutral molecule, which is the most frequent type of collision occurring in collisional plasmas, where the interaction time between the two particles is on the order of 10−16 s, and the effect of the external force due to an imposed electric field is negligible compared with the huge internal force between the two particles. The conservation of

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

48

momentum is then simply given by mv′ + MV ′ = mv + MV,

(4.14)

where (m, M) are, respectively, the masses of the charged and neutral particle, and (v′ , V ′ ) are the velocities before and (v, V) the velocities after the collision. A similar conservation law is written for the energy. However, in this case internal excitation can occur and the inelastic process should also be accounted for. In the energy conservation law, it is convenient to change the frame of reference to a center-of-mass (CM) frame, and the kinetic energy is represented as εkin =

1 2 1 1 1 mv + MV 2 = (m + M)vg2 + µr vr2 , 2 2 2 2

(4.15)

where the CM velocity vg and the relative velocity vr are given as vg =

mv + MV m+M

(4.16)

and vr = v − V.

(4.17)

The CM velocity remains constant before and after the collision under the conservation of momentum expressed in Equation 4.14. As a result, the transformation from the standard laboratory (LAB) system to the CM system effectively allows us to describe a two-particle process as a single-particle process. The reduced mass of the effective particle is therefore µr =

mM . m+M

(4.18)

Using these transformations to the CM frame, we can write the energy conservation law as 1 1 2 µr vr′ = µr vr2 + ε R , 2 2

(4.19)

where ε R is the energy loss (reaction energy) due to the excitation of internal energy levels of one or both particles. We define three kinds of collisions, each defined by their value and sign of the reaction energy. i. ε R > 0: Collisions of the first kind, in which we have loss of the total kinetic energy to inelastic processes; ii. ε R < 0: Collisions of the second kind, in which we have gain of the total kinetic energy from the internal excitation energy; and iii. ε R = 0: Elastic collisions, in which the total kinetic energy is conserved.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

49

PROBLEM 4.2.1 Explain that the CM energy is conserved in the case of binary collision, and that in the case of inelastic scattering the energy to excite the molecule is provided from the relative kinetic energy. 4.2.2 Definition of Collision Cross Sections The concept of a collision cross section can be best understood when we consider a collision between two particles represented as hard spheres with radii of a 1 and a 2 , approaching with a relative velocity vr along the z-axis. A hard sphere is an object for which the interaction potential is zero for distance r > a 0 and infinitely repulsive for r ≤ a 0 . We now introduce the impact parameter b to the binary collision with b representing the perpendicular distance from the z-axis along the initial relative velocity to the center of the target. We then observe (Figure 4.5) that only those particles with an impact parameter b less than a 1 + a 2 will collide, whereas others will continue their motion without changing their direction. Thus the circle with radius a 0 = a 1 + a 2 is the effective size of the target for projectiles represented as points. The area of this circle πa 02 = π(a 1 + a 2 )2 is the effective collision cross section for the two hard spheres. PROBLEM 4.2.2 Derive the energy transfer as a function of scattering angle in an elastic collision of two hard spheres (r, m, v′ ) and (R, M, 0), both defined by their radius, mass, and velocity before collision. The concept of an “effective area” associated with target particles for any pair of colliding particles (two-body collision) is useful even if the hard sphere approximation is not valid. There are, however, some conditions that must be met for it to be applicable. i. The collisions between incident particles should be negligible; that is, the density of the incident particles should be sufficiently low;

a2 a1

v V

a0 z

v

FIGURE 4.5 Binary encounter of solid spheres with radii of a 1 and a 2 , and the concept of the collision cross section.

© 2006 by Taylor & Francis Group, LLC

50

Plasma Electronics: Applications in Microelectronic Device Fabrication ii. The density of the target particles should be low in order to avoid collisions of the incident particles with more than one target at the same time. That is, the de Broglie wavelength of the incident particle should be shorter than the mean distance between target molecules.

In defining the collision cross section we return to the differential cross section that was mentioned earlier. When a certain number of projectiles n with energy ε are directed toward a target consisting of N particles, the number of particles dn scattered into the surface element dS on the spherical surface at a distance r from the target center will be proportional to the density of the projectiles and the targets and also proportional to the area of the surface dS and r −2 , that is, the solid angle in the collision experiment: dn ∝ nN ds/r 2 , where ds/r 2 = d(= sin θdθ dφ). It is known that dn depends on the incident energy of the projectile and the quantum property of the target molecule. If we define the constant of proportionality as σ (θ, φ; ε), then the relation above becomes dn = σ (θ, φ; ε)nNd.

(4.20)

In Equation 4.20, σ (θ, φ; ε)d is the probability that one projectile is scattered into a small solid angle d at scattering angle (θ, φ). σ (θ, φ; ε) has the dimensions of the area and so is termed the “differential cross section.” The differential cross section may be expanded into the Legendre polynomials Pn (cos θ ) under the azimuthal symmetry σ (θ; ε) =

∞ 1 (2n + 1)Qn (ε)Pn (cos θ), 4π 0

(4.21)

where Qn (ε) is the nth integral cross section. Multiplying the equation by Pn (cos θ ) and using the property of orthogonality of the Legendre polynomials (see Section 5.6.1), we obtain the following definition of the integral cross section Qn (ε): π Qn (ε) = 2π σ (θ; ε)Pn (cos θ) sin θdθ. (4.22) 0

The coefficients Qn (ε) and the differential cross section σ (θ ; ε) are defined for each of the many scattering processes that can occur. They are interpreted as the scattering probability averaged over all angles and weighted by some Legendre polynomial. For example, Q0 (ε) is the total cross section for the process (i.e., the probability of scattering at any angle) and corresponds to the area of the target as seen by a point projectile. The most commonly used angular averages of the differential cross section are the following:

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces •

Total cross section Q0 (ε) =

•

51

σ (θ, φ; ε)d = 2π

π

σ (θ; ε) sin θdθ;

(4.23)

0

Momentum transfer cross section Qm (ε)(= Q0 − Q1 ) = (1 − cos θ )σ (θ; ε)d π = 2π (1 − cos θ )σ (θ; ε) sin θdθ; and

(4.24)

0

•

Viscosity cross section 2 Qv (ε) = (Q0 − Q2 ) = (1 − cos2 θ)σ (θ; ε)d 3 π (1 − cos2 θ)σ (θ; ε) sin θdθ. = 2π

(4.25)

0

These integral forms of differential cross section play a special role in transport theory. For inelastic collisions with threshold energy ε j , the definitions are the same except that the momentum transfer cross section is defined as π εj 1− 1− cos θ σ (θ; ε) sin θdθ, (4.26) Qm (ε) = 2π ε 0 > ε j when the scattering is nearly isotropic, Qm (ε) will be very Thus, at ε ∼ small.

Exercise 4.2.1 Explain the physical meaning of the momentum transfer cross section for elastic scattering. Assuming that the initial momentum is p and that particles are scattered at an angle θ with no loss of the magnitude of the velocity, the difference of momentum along the axis of the initial velocity is p = p − p cos θ = p(1 − cos θ). In other words, the fractional change of the momentum is p = (1 − cos θ). p

(4.27)

Therefore, the cross section Qm in Equation 4.24 is in some way a representation of the momentum transfer to the gas molecules. Qm for elastic collision has a finite magnitude, and the elastic energy loss (2m/M)ε of the electron with energy ε is negligibly small. PROBLEM 4.2.3 Discuss the physical meaning of the viscosity cross section.

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

52

Exercise 4.2.2 Gas molecules with density N and temperature Tg are present in a reactor. Estimate the limitations of the two-body approximation for the collisions between molecules. This criterion may be related through the relationship between the mean distance between molecules d and the de Broglie wavelength λde of the molecule, d ≫ λde . The mean distance between the gas molecules may be obtained from the gas number density N as d = N−1/3 . On the other hand, the wavelength, which is used to approximate the range over which quantum effects are appreciable, will be calculated for the most probable speed of molecules, 2kTg /M. Hence the de Broglie wavelength is given by λde =

h h h . ≈ = p M˜v 2MkTg

The condition therefore becomes

h N1/3 λde ≈ ≪ 1. d 2MkTg

(4.28)

PROBLEM 4.2.4 Calculate Q0 , Qm , and Qv for the following models of differential scattering cross sections: •

Isotropic (σ (θ) is independent of angle);

•

Forward (σ (θ) is a delta function at zero angle);

Backward (σ (θ) is a delta function at π ); and • Right angles (σ (θ) is a delta function at π/2).

•

Discuss the relative magnitudes Qm /Q0 and Qv /Q0 in these cases. The unit usually used in presenting a collision cross section between an electron and a molecule is 10−16 cm2 . This unit originates from the Angstrom ˚ an old unit of length equal to 10−8 cm (0.1 nm). Other units are often (A) applied as well. For example, it is convenient to represent the cross section by using the Bohr radius (i.e., the first Bohr orbit of H), a 0 = 0.529 × 10−8 cm as [a 02 ] = 0.280 × 10−16 cm2

4.29a )

The effective cross section of a Bohr radius is then given as [πa 02 ] = 0.880 × 10−16 cm2

4.29b)

In the case of photon scattering, it is common to use the “barn” [barn] = 10−24 cm2

© 2006 by Taylor & Francis Group, LLC

4.29c)

Elementary Processes in Gas Phase and on Surfaces

f (ε)

1 exp λ

( λz ) z

0 ε

53

collision

electron beam

FIGURE 4.6 Distribution of the mean free path f (z; ε) for a beam of particles with energy ε incident on a gas target with random distribution.

Finally, we note that the total collision probability Pc is often used to represent the number of collisions in a gas at some pressure p and at temperature Tg [K]. The total collision probability is equal to Pc = ( p/kTg )Q. Here, the standard value that is used is p = 1Torr at room temperature, where we have Pc = 9.66 × 1018 Q[cm2 ]/Tg [K]cm−1 Torr−1

(4.29d)

for the standard collision probability at 1 Torr = 133Pa . 4.2.3 The Distribution of Free Paths In order to obtain the distribution of the free paths of a particle between collisions, we consider a beam of particles with density n0 and energy ε entering gas at z = 0 (see Figure 4.6). It can be shown that the distribution function of the free path is given by z 1 exp − , (4.30) f (z; ε) = < λ(ε) > < λ(ε) > where < λ(ε) > is the mean free path of a particle with energy ε.

Exercise 4.2.3 Derive the distribution function of the free path (Equation 4.30). When we define F (z) as the integral distribution of collisions on the segment (0, z), the number of collisions in the small segment (z, z + z) is proportional

© 2006 by Taylor & Francis Group, LLC

54

Plasma Electronics: Applications in Microelectronic Device Fabrication

to the number of particles remaining in the beam (1 − F (z)) and to dz, so we have

F (z + dz) − F (z) = c{1 − F (z)}dz,

where c is a constant. Next we obtain the equation

F (z + dz) − F (z) −1 = −c, 1 − F (z) dz which has a solution ln{1 − F (z)} = −cz or

F (z) = 1 − exp(−cz).

From the integral distribution of collisions F(z) we determine the distribution of the free path between collisions: f (z) =

dF (z) = c exp(−cz). dz

Considering the definition of the mean free path, we have ∞ zf (z)dz 1 = , < λ > = 0∞ c f (z)dz 0

and thus the formula (Equation 4.30) for the distribution function of the free path is obtained. The distribution f (z) can be used to determine the probability of collision (or of the length of the free path) of a single particle as well. One should bear in mind that the mean free path depends on the energy of the incident particles, that is, < λ(ε) >. Thus if we have a distribution of incident energies we perform additional averaging and obtain the mean free path for the whole ensemble. 4.2.4 Representation of Collisions in Laboratory and CM Reference Frames In Section 4.2.1, we have shown that the motion of two particles of masses m and M and velocities v and V can be treated as a single-particle motion with relative mass µr and relative velocity vr by considering the total energy conservation εkin =

1 1 1 2 1 mv + MV 2 = (m + M)vg2 + µr vr2 , 2 2 2 2

(4.31)

where vg is the CM velocity. A convenient way to describe a binary collision is to make a transfer from the laboratory frame of reference to the CM frame of reference of the relative mass µr and the velocity vr (see Figure 4.7).

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

55

v' π−ω c 2

v

ωc vg

ωL

V V'

FIGURE 4.7 Velocities of two particles before and after collision and the corresponding angles in the CM frame.

Exercise 4.2.4 Obtain the relationship between differential cross sections for the laboratory and the CM reference frames in the elastic scattering. Consider two particles, (m) and (M), moving toward one another with velocities (v′ , V ′ ) before and (v, V) after the collision in the LAB frame. The magnitude of the relative velocity vr is conserved before and after the elastic scattering in the CM frame, (| v − V |=| v′ − V ′ |=| vr |). We may thus represent the motion of particles before and after the scattering by two straight lines of equal lengths (|vr |) intersecting at the point vg (CM velocity) (see Figure 4.7). In the CM frame, the velocities of the two particles before and after the collision are located on spheres with radii of v = vr M/(m + M) and V = vr m/(m + M), respectively. We can identify one equilateral triangle with the CM frame scattering angle ωc , that is, the triangle [vg , v′ , v]. Thus we write Mvr /(m + M) v′ v = , sin ωc sin[(π − ωc )/2] where the length between points [v, v′ ] is denoted v′ v. If, however, we take the triangle [V ′ , v′ , v], the following relation is obtained by using the LAB frame scattering angle of particle m, ω L : vr v′ v = . sin ω L sin[π − ω L − (π − ωc )/2]

© 2006 by Taylor & Francis Group, LLC

56

Plasma Electronics: Applications in Microelectronic Device Fabrication

Combining the results from the two triangles we obtain tan ω L =

sin ωc (m/M) + cos ωc

(4.32a )

and cos ω L =

(m/M) + cos ωc

1 + 2(m/M) cos ωc + (m/M)2

1/2 .

(4.32b)

Because the same number of particles is scattered through the solid angle in both systems, we have σ L (ω L ; ε)d L ≡ σc (ωc ; ε)dc , or d cos ωc d cos ω L 3/2

1 + 2(m/M) cos ωc + (m/M)2 . = σc (ωc ; ε) 1 + (m/M) cos ωc

σ L (ω L ; ε) = σc (ωc ; ε)

(4.33)

It is possible to show that the energy transfer from the projectile to the target (i.e., the energy loss to the first particle) is then equal to 2 MV 2 /2 2mv ′ M εM = = cos ω L ′ εm mv ′2 /2 mv ′ 2 m + M 4mM cos2 ω L . = (m + M)2

(4.34)

PROBLEM 4.2.5 Determine the scattering angles ω L and ωC M when (a) m = M and (b) m ≪ M. PROBLEM 4.2.6 Show that the energy loss in elastic collisions is given by Equation 4.34, and that for the electron-molecule scattering it is simplified after averaging to (2m/M)ε. PROBLEM 4.2.7 Prove that for a head-on (ω L = 0) collision of a heavy particle with a light target m ≫ M, the maximum velocity of the target will be V = 2v ′ . It is also of interest to determine the velocities after the collision in the case of inelastic processes characterized by the inelastic energy transfer ε. After inelastic scattering, the velocity of the target in the LAB frame is equal to

1 m ′ v= [mMv ′ 2 cos 2 θ L − 2(m + M)ε] (4.35) mv cos θ L ± m+M M

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

57

and the velocity of the projectile is 1 M ′ 2 V= mv cos ω L ± v ′ (M2 − m2 sin2 ω L ) − 2(m + M) ε , m+M m (4.36) where θ L is the scattering angle of the target (M) in the LAB frame. From Equations 4.35 and 4.36 it is possible to obtain the minimum velocity of the incident particle necessary to achieve inelastic excitation with a loss ε: 1 ′2 (m + M)M mv = ε, 2 M2 − m2 sin2 ω L

(4.37)

which for head-on collisions reduces to ′ εm =

m+M ε. M

′ For particles of the same mass (m = M) one needs at least εm = 2ε to achieve an inelastic process, whereas for light projectiles (m ≪ M) the ′ minimum energy is εm = ε.

4.3 Classical Collision Theory In the classical theory of collisions it is possible to determine the differential cross section based on the interaction potential and conservation laws in a binary collision. It is always useful to reduce the problem of scattering of two particles to the scattering of one particle with a reduced mass. It is assumed that the target molecule has a centrifugal potential that permits spherical symmetry in the scattering event. With the exception of the head-on collision, the incoming particle will pass the target at some distance. The point of the closest approach is known as the impact parameter (b). Even in the case when there is an external force field to deflect the particle, we may define the instantaneous impact parameter for any point as a function of the spherical coordinates that describe the motion. It is evident that impact parameter b will define the degree of interaction and final trajectory for a given interaction potential. It is also clear that the outcome of scattering is described by the angle of scatering θ. In Figure 4.8 we show several classical trajectories for a projectile scattered from a target molecule. There, we define the scattering angle θ and the corresponding annular solid angle. It is also obvious that, in general, the smaller the impact parameter, the greater the scattering angle, because the particle will approach close to the target and thus be subjected to a stronger interaction. There is a unique relation between the scattering angle and the impact parameter b = b(θ) (note this may not be true in some special circumstances, e.g., when orbiting occurs).

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

58

v0

v0

dθ db

θ

b

z 0

2πbdb

dΩ = 2πsinθdθ

FIGURE 4.8 Definition of the scattering angles and the impact parameter in classical scattering phenomenology.

The number of particles scattered into the angle θ is proportional to the solid angle d and the differential cross section σ (θ). The scattered particles with probability σ (θ)d correspond to the particles incident on the area of the annular surface 2π bdb. Therefore, σ (θ) = 2π bdb/d db(θ) dθ = b(θ) dθ sin θdθ db(θ) 1 . b(θ) = sin θ dθ

(4.38)

Our goal is thus to determine the dependence b(θ) for any given potential interaction. 4.3.1 Scattering in Classical Mechanics We now proceed to calculate the function b(θ) for scattering on a target placed at the scattering center r = 0. The projectile has velocity v and reduced mass µ. The interaction potential is V(r ), which is assumed to be spherically symmetric (i.e., a centrifugal potential). Additionally, the scattering is assumed to be purely elastic. Because we must satisfy the basic laws of conservation, the conservation of angular momentum L yields L = µv0 b = µ(r dξ/dt)r = µv0′ b ′ .

© 2006 by Taylor & Francis Group, LLC

(4.39)

Elementary Processes in Gas Phase and on Surfaces

59 v0

1 (π − θ) 2

1 (π − θ) 2 v0′

r

rmin

b θ

ξ

b′

FIGURE 4.9 The geometry of classical scattering on a target with interaction potential V(r ).

The conservation of the total energy E yields E = µv02 /2 = µ{(dr/dt)2 + r 2 (dξ/dt)2 }/2 + V(r ) 2

= µv0′ /2,

(4.40)

where ξ is defined as in Figure 4.9. At positions sufficiently far from the scattering center, Equations 4.39 and 4.40 give the relations of the velocities and the impact parameters before and after collision: v0 = v0′ ,

b = b′.

(4.41)

In Figure 4.9 we define the coordinates for the scattering and basically the incoming particle with reduced mass µ follows the angle ξ . From Equation 4.39 it is shown that dξ/dt = L/µr 2 ,

dr/dt = (dξ/dt)(dr/dξ ) = (L/µr 2 )(dr/dξ ),

and from Equation 4.40 it follows that E = µ{(L/µr 2 )2 (dr/dξ ) + r 2 (L/µr 2 )2 }/2 + V(r ) = {L 2 /2µr 4 }{(dr/dξ )2 + r 2 } + V(r ). The equation describing the particle trajectory is therefore (dr/dξ )2 = {2µr 4 /L 2 }{E − V(r )} − r 2 .

(4.42)

It is also worth noting that µv 2 b 2 L2 + V(r ) = + V(r ) = Veff (r ), 2µr 2 2r 2

(4.43)

and Veff (r ) is the effective potential in the radial direction. The interaction potential usually has a minimum; as a result, the projectile is for a while accelerated to the target, but it will not be bound as it has a positive energy. The addition of the centrifugal term to the interaction potential leads to an increase in the potential in the region of the minimum as the centrifugal term

© 2006 by Taylor & Francis Group, LLC

60

Plasma Electronics: Applications in Microelectronic Device Fabrication

increases. The potential may even become repulsive. It may also be possible that a maximum is formed at some distance rc with a shallow minimum for smaller distances. Near the maximum the radial motion will be very small, but it will take several revolutions before the particle can leave after scattering as the angular momentum is large. This effect is known as “orbiting.” Orbiting allows longer interaction times and therefore a greater probability for processes that involve transitions with finite lifetimes. To calculate the trajectory from Equation 4.43, we must obtain an equation that will describe the trajectory through the angle ξ , dξ = ± dr 2µr 4 L2

1 {E − V(r )} − r 2

1/2 .

(4.44)

When dξ/dr < 0, the incoming particle approaches the target and the interaction force increases. When dξ/dr > 0, the projectile leaves the target. Because of the symmetry of the scattering (see Figure 4.9), we may perform the integration in two parts. First we perform it up to the point of the closest approach rmin , from ξ = 0 (for r → ∞) up to ξ = (π − θ )/2. We obtain

(π−θ)/2

0

dξ =

(π − θ )/2 =

rmin

∞∞ rmin

−[{2µr 4 /L 2 }{E − V(r )} − r 2 ]−1/2 dr

[{2µr 4 /L 2 }{E − V(r )} − r 2 ]1/2 dr.

(4.45)

The limit rmin (the point of the closest approach) may be obtained from 4 dr/dξ = 0, which in combination with Equation 4.45 leads to {2µrmin /L 2 } 2 {E − V(rmin )} − rmin = 0. Introducing u = 1/r , we derive the following result from Equation 4.45: umax (π − θ )/2 = [{2µ/L 2 }{E − V(1/u) − u2 }]−1/2 du; 0

(2µ/L 2 ){E − V(1/umax )} − u2max = 0.

(4.46)

An analogous equation may be obtained using the impact parameter b: umax (π − θ )/2 = [(1/b 2 ){1 − V(1/u)/E} − u2 ]−1/2 du; 0

(4.47) 1 − V(1/umax )/E − b 2 u2max = 0.

Exercise 4.3.1 Determine the dependence of the impact parameter b(θ) for classical hard ≤ r0 ) . Also detersphere elastic scattering, that is, for the potential V(r ) = ∞(r 0(r > r0 ) mine the effective cross section for the scattering. In Figure 4.10 we show that the hard sphere potential is a very good approximation for many classical potentials. The point of minimum approach

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

61

∞

V(r)

hard sphere realistic potential

r

r0

0 π−θ

2

π−θ

2 π−θ

2 b (θ) r0

0

FIGURE 4.10 Comparison between the hard sphere interaction potential and a realistic potential (dashed line) and geometry of the hard sphere scattering for a given impact parameter.

is rmin = r0 = 1/umax and may be used in Equation 4.47 to obtain (π − θ)/2 =

0

1/r0

[(1/b 2 ) − u2 ]−1/2 du 1/r0

= sin−1 [u/(1/b)]0

= sin−1 [b/r0 ]; (b ≤ r0 ) = no collision; (b > r0 ). We obtain the solution, sin[(π − θ)/2] = b/r0 ; that is, θ (b) = 2 cos−1 (b/r0 ) (b < r0 ) = 0 (b > r0 ). Substituting the above relation into Equation 4.38 we obtain the expression for the differential cross section: σ (θ) = (1/ sin θ )b(θ)|db(θ)/dθ| = (1/ sin θ )r0 cos(θ/2)| − (r0 /2) sin(θ/2)| = r02 sin θ/(4 sin θ) = r02 /4,

© 2006 by Taylor & Francis Group, LLC

(4.48)

62

Plasma Electronics: Applications in Microelectronic Device Fabrication

and from the definition of the total cross section Equation 4.23 we obtain Q0 = σ (θ)d = (r02 /4) sin θ dθdφ = (r02 /4)4π = πr02 .

(4.49)

PROBLEM 4.3.1 Derive the momentum transfer cross section Qm for hard sphere elastic scattering, and discuss the relationship between momentum transfer and the total cross section.

Exercise 4.3.2 Derive the impact parameter dependence of the scattering angle θ(b) and of the differential cross section σ (θ) for a light H+ (with charge e and mass m) and massive ion (with charge Ze and mass M) in a fully ionized plasma (i.e., collisionless plasma) by considering the classical Rutherford scattering in a Coulomb field. Classical Rutherford scattering and its physical quantities are shown in Figure 4.11. We may regard Coulomb scattering as resulting in small angle scattering or small changes of the momentum under the long-range interaction. Thus it is reasonable to assume that sin θ ∼ θ = pT / p. Here, the Coulomb force will induce a velocity perpendicular to the original velocity and the resulting momentum is denoted by pT . We may calculate the transverse momentum by integrating the effect of the force FT in the region of interaction ∞ ∞ (Ze 2 /4π ε0r 2 ) cos βdt FT dt = pT = −∞ −∞ ∞ = {Ze 2 /4π ε0 (b 2 + v 2 t 2 )}{b/(b 2 + v 2 t 2 )1/2 }dt −∞

= 2Ze 2 /4πε0 bv. A

vt

e p = mv b

e

θ

pT

β r

Ze FIGURE 4.11 Classical Rutherford scattering between a light H+ (with charge e and mass m) and a massive ion (with charge Ze and mass M) in a Coulomb field.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

63

Therefore, θ (b) = pT / p 2Ze 2 1 2Ze 2 = · = , 4π ε0 bv mv 4π ε0 mv 2 b

(4.50)

and the differential cross section can be obtained from Equation 4.38 as σ (θ) = (1/ sin θ )b(θ)|db(θ)/dθ| 1 2Ze 2 2Ze 2 −1 ∼ θ 4π ε0 mv 2 θ 4π ε0 mv 2 θ 2 2 4 Ze 2 . ∼ 2 4π ε0 mv θ4

(4.51)

A more detailed analysis gives θ(b) and σ (θ) for the Rutherford scattering: Ze 2 −1 ; θ (b) = 2 tan 4π ε0 mv 2 b 2 Ze 2 1 1 . (4.52) σ (θ) = 4 4π ε0 mv 2 sin4 (θ/2) It appears that it is not possible to integrate the differential cross section of the Rutherford scattering into total or momentum transfer cross sections as there is a singularity for θ ∼ 0. However, we should remember that the long-range effect of the Coulomb force will cancel out in ionized gas due to the effect of other charged particles. The effect of the Coulomb interaction is considered only within the so-called Debye radius λ D ; that is, we need to consider the interaction only at b < λ D with the minimum scattering angle θmin at b = λ D . In other words, there is a limit on impact parameter b ≤ b max . The momentum transfer cross section is therefore 2 π 1 1 Ze 2 Qm = (1 − cos θ )2π sin θ dθ 4 2 4πε0 mv sin (θ/2) θmin 4 2 2 Ze 2 . ln = 2π 4πε0 mv 2 1 − cos θmin Here, 1 − cos θ = 2/{1 + [b/(Ze 2 /4π ε0 mv 2 )]2 } in Figure 4.11, and we may determine the cross section Qm by using 1 − cos θmin = 2[(Ze 2 /4π ε0 mv 2 )/b max ]2 ,

where b max = λ D .

We thus obtain Qm = 4π(Ze 2 /4π ε0 mv 2 )2 ln[4π ε0 mv 2 λ D /Ze 2 ] Z2 e 4 = ln . 4πε02 (mv 2 )2

© 2006 by Taylor & Francis Group, LLC

(4.53)

64

Plasma Electronics: Applications in Microelectronic Device Fabrication

Here ln is defined as ln = ln

4π ε0 mv 2 λ D Ze 2

(4.54)

and is known as the Coulomb logarithm. PROBLEM 4.3.2 Show that the Coulomb logarithm corresponds to the ratio b max /b min . 4.3.2 Conditions for the Applicability of the Classical Scattering Theory Two conditions must be met to allow us to use the classical scattering theory: i. The trajectory of the particle must be clearly defined. In other words, the de Broglie wavelength λde must be much shorter than the effective range of the interaction potential d0 , or λde = (h/µvr ) ≪ d0 .

(4.55)

ii. The change of the momentum p must be much greater than the uncertainty pd ≫ h. For an effective range of the potential V0 , the conservation of energy gives vr p ∼ eV0 . The condition is thus eV0 d/vr ≫ h.

(4.56)

4.4 Quantum Theory of Scattering Collisions are considered in quantum theory under three major conditions: (a) The incident particle is described by a plane wave. (b) The total energy of the incident particle is arbitrary positive. (c) The scattered particle is analyzed at a point far from the scattering center, where the influence of the interaction potential is negligible. We discuss the elastic scattering of a particle with energy E. A schematic diagram of the scattering is shown in Figure 4.12, where the region is divided into three points (a), (b), and (c). (a) Incident wave. In this subsection, the incoming particle is represented as the wave function (r, t), which can be written as (r, t) = exp[i(k · r − ωt)],

(4.57)

where p = h¯ k and E = h¯ ω. The description in Equation 4.57 is valid only at a long distance where the interaction potential is negligible.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

65 f (θ, φ) ikz r e (spherical wave)

π−θ V(r) Aeikz (plane wave) FIGURE 4.12 The planar wave of the incoming particle and the spherical wave of the scattered particle in quantum theory of elastic scattering.

When the velocity of the incoming particle is along the z-direction in the steady state, then (z) = exp(ikz).

(4.58)

(b) Under scattering potential. Here, the collision is characterized by the Schrodinger ¨ equation having a continuous positive energy E of the incoming particle and the interaction potential V(r ). Hence the wave equation reads as −(¯h 2 /2µ)∇ 2 (r) + V(r)(r) = E(r).

(4.59)

(c) After scattering. From the condition that scattered particles eventually escape to far distances unaffected by V(r ) and distribute themselves spherically around the scattering center (target), we may conclude that the scattered particle can be described by a spherical wave. Thus, the total wave function is the sum of the scattered spherical wave and the unaffected plane wave: (r) = exp(ikz) +

f (k; θ, φ) exp(ikr ). r

(4.60)

The spherical term is normalized by f (k; θ, φ), which is known as the scattering amplitude. For example, if there is no interaction, the scattering amplitude is zero and the planar wave is unaffected. The number of particles n scattered into a small area dS normalized to the total number of the incident particles n represents the probability of the scattering in a given direction (θ, φ) and is therefore equal to 2 f (k; θ, φ) 1 n/n = exp(ikr ) dS · exp(ikz)2 r = | f (k; θ, φ)|2

© 2006 by Taylor & Francis Group, LLC

dS . r2

66

Plasma Electronics: Applications in Microelectronic Device Fabrication

Here, the solid angle is d = dS/r 2 , and by definition of the differential cross section σ , we have n/n = σ (k; θ, φ)d. We can then obtain the relation between the differential cross section and the scattering amplitude in quantum theory as σ (k; θ, φ) = | f (k; θ, φ)|2 .

(4.61)

4.4.1 Differential Scattering Cross Section σ(θ) We can solve Equation 4.59 by using Equation 4.60, making a solution in the CM system for a stationary target and projectile with a reduced mass µ. The wave function can be separated into two terms, the radial (r ) and angular (θ, φ) functions; that is, (r ) = R(r )Y(θ, φ). In the case of azimuthal symmetry, it is useful to use Legendre polynomials (and take advantage of their orthogonality): (r) = R(r )Pl (cos θ ).

(4.62)

This makes it possible to derive the differential equation for the radial part of the wave function R(r ): h¯ 2 l(l + 1) 2µE 2µ 1 d 2 dR(r ) R(r ) = 0. (4.63) r + − 2 V(r ) + r 2 dr dr 2µ r 2 h¯ 2 h¯ Here, we have used quantization of the angular momentum. We define the effective radial potential as Veffl (r ) = V(r ) +

h¯ 2 l(l + 1) . 2µ r 2

(4.64)

As in the classical case, the effective radial potential is the result of the combined effect of the interaction potential and the centrifugal term. When we make transformation R(r ) = u(r )/r , the differential equation can be written as −

h¯ 2 d 2 u(r ) + Veffl (r )u(r ) = Eu(r ). 2µ dr2

(4.65)

We now consider the asymptotic behavior of the solutions. From Veffl → 0 when r → ∞ it follows that u(r ) → Asin(kr ) + B cos(kr ) → Asin(kr + ηl − lπ/2); at r → ∞.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

67

The radial component of the wave function has the effect of scattering through a phase-shift (ηl − lπ/2) at a long distance from the scattering center (target). It should be noted that without scattering u(r ) is exactly equal to the incoming plane wave, expressed below as Asin(kr −lπ/2). Consequently, the solution for R(r ) is R(r ) = (1/kr ) sin(kr − lπ/2 + ηl )

at r → ∞,

(4.66)

and by combining Equations 4.59, 4.62, and 4.66, we obtain the general solution for the wave function as Al R(r )Pl (cos θ ) (r) = l

∼ ∼

Al (1/kr ) sin(kr − lπ/2 + ηl )Pl (cos θ)

l

(at r → ∞)

Al (1/2ikr ){exp[i(kr − lπ/2 + ηl )]

l

− exp[−i(kr − lπ/2 + ηl )]}Pl (cos θ)

(at r → ∞).

We must now match the asymptotic forms of the above general solution and Equation 4.60. For that purpose we must expand the incoming plane wave, exp(ikz) in Equation 4.60 for r → ∞ as exp(ikz) = exp(ikr cos θ ) (2l + 1) 1 = exp(ikrt)Pl (t)dt t = cos θ Pl (cos θ ) 2 −1 l

(Rayleigh formula)

=

Pl (cos θ )

l

2l + 1 l 2i jl (kr ). 2

Recall some of the mathematical relations described above, including the orthogonal functions f (x) =

l

(2l + 1) a l Pl (x) and a l = 2

1

f (t)Pl (t)dt,

(4.67)

−1

the integral expression of the spherical Bessel function jl (x) = (1/2i l )

1

exp(i xt)Pl (t)dt,

(4.68)

−1

and the asymptotic form of the spherical Bessel function in the limit x → ∞: jl (x) = (π/2xl )1/2 J l+1/2 (x) ∼ (1/x) cos (x − (2l + 1)π/2)

© 2006 by Taylor & Francis Group, LLC

(at x → ∞).

Plasma Electronics: Applications in Microelectronic Device Fabrication

68

Thus, we may write the asymptotic expansion of the planar wave in the limit r → ∞ as exp(ikz)r →∞ (2l + 1)i l (1/kr ) sin(kr − lπ/2)Pl (cos θ) ∼ l

(2l + 1)i l lπ lπ exp i kr − − exp −i kr − Pl (cos θ ) ∼ 2ikr 2 2 l

(4.69)

As a result, the scattering wave is obtained from (r) − exp(ikz) 2l + 1 l Pl (cos θ ) Al R(r )Pl (cos θ ) − 2i jl (kr ) = 2 l l=0 1 lπ lπ ∼ Al + ηl − exp −i kr − + ηl Pl (cos θ ) exp i kr − 2ikr 2 2 l=0 (2l + 1)i l lπ lπ − exp i kr − − exp −i kr − Pl (cos θ) 2ikr 2 2 l 1 lπ Pl (cos θ ) exp i kr − Al exp(iηl ) − (2l + 1)i l ∼ 2ikr 2 l=0 1 lπ − exp −i kr − [Al exp(−iηl ) − (2l + 1)i l ] (at r → ∞). (4.70) 2ikr 2 Equation 4.70 should be matched to the outgoing spherical wave exp(ikr )/r . Therefore, the second term in Equation 4.70 must satisfy [Al exp(−iηl ) − (2l + 1)i l ] = 0,

i.e., Al = (2l + 1)i l exp(iηl ).

The asymptotic form of the scattered spherical wave is f (k; θ ) exp(ikr ) r 1 lπ exp i kr − [(2l + 1)i l exp(2iηl ) − (2l + 1)i l ]Pl (cos θ) ∼ 2ikr 2 l=0 1 ilπ l exp(ikr ) ∼ exp − , (4.71) i (2l + 1){exp(2iηl ) − 1}Pl (cos θ) 2ik 2 r l=0 and from the expression for f (k; θ), it is possible to obtain the differential cross section σ (θ): f (k; θ ) =

© 2006 by Taylor & Francis Group, LLC

1 (2l + 1){exp(2iηl ) − 1}Pl (cos θ) 2ik l=0

(4.72)

Elementary Processes in Gas Phase and on Surfaces

69

u(r) V(r) = 0 V(r) > 0 r

kr0 ηl FIGURE 4.13 The effect of phase-shift on the wave function.

σ (θ) = | f (k; θ )|2 2 1 2l + 1 = 2 {exp(2iηl ) − 1}Pl (cos θ) . k 2

(4.73)

l=0

Hence the only quantities that describe the effect of the scattering are the phase-shifts ηl . The use of angular momentum quantum numbers l is analogous to the use of an impact parameter in classical scattering. The scattering cross section is obtained by summing up the different partial waves in respect to l = 0, 1, 2, . . . , and the procedure shown here is known as the partial wave method. This technique was developed for light scattering by Rayleigh and applied to particle scattering by Faxen and Holtsmark. PROBLEM 4.4.1 Show that the total cross section Q0 of the elastic scattering is given in partial wave expansion as Q0 (ε) = | f (k; θ)|d (4.74) =

4π (2l + 1) sin2 ηl . k 2 l=0

Exercise 4.4.1 Explain the phase difference ηl in Equation 4.66. In Figure 4.13, we show two functions u(r ), one without any interaction with a target (V(r ) = 0; solid line) and one with an interaction (V(r ) > 0; dashed line). When the potential is repulsive, the outgoing wave function is pushed to the outside as compared with the wave at V(r ) = 0, and then ηl > 0. On the other hand, if it is attractive, then ηl < 0.

Exercise 4.4.2 In Figure 4.14 we show the spherical well potential: V(r ) = −V0

r ≤ r0 = 0 r > r0 .

© 2006 by Taylor & Francis Group, LLC

70

Plasma Electronics: Applications in Microelectronic Device Fabrication

V(r) 0

r0

r

-V0

FIGURE 4.14 The spherical well potential (spherical attractive potential).

Calculate the total cross section and the phase-shifts for elastic scattering of a low-energy particle. From Equation 4.65, we have d 2 u(r )

2µ l(l + 1)u(r ) = 0, {E − V(r )}u(r ) − 2 r2 h ¯ dr and for the potential it follows that 2

+

< {d 2 /dr2 + K 2 − l(l + 1)/r 2 }ul,k (r ) = 0; K 2 = (2µ/¯h 2 )(E + V0 ); 2

2

2

{d /dr + k − l(l + 1)/r

2

> }ul,k (r )

2

2

= 0; k = (2µ/¯h )E;

(r ≤ r0 ), (r > r0 ).

The solution for R(r ) = u(r )/r must be finite at r = 0, and the u(r ) must be zero at r = 0. For low-energy scattering that satisfies K 2 − l(l + 1)/r 2 > 0, the component in the partial wave expansion will mainly be l = 0 (s-wave). Therefore, u0,k (r ) = u< 0,k (r ) = Asin K r = u> 0,k (r ) = B sin(kr + η0 ). The wave function u(r ) and its derivative satisfying the Schrodinger ¨ equation must be continuous at r = r0 . Then, Asin K r0 = B sin(kr0 + η0 ), K Acos K r0 = k B cos(kr0 + η0 ), and we obtain (1/K ) tan(K r0 ) = (1/k) tan(kr0 + η0 ), k (or) η0 = tan−1 tan(K r0 ) − kr0 . K As a result, the scattering cross section for l = 0 (s-wave) is

4π k 4π tan K r0 − kr0 . Q0 (k) = 2 sin2 η0 = 2 sin2 tan−1 k k K

© 2006 by Taylor & Francis Group, LLC

(4.75)

Elementary Processes in Gas Phase and on Surfaces

71

V(r) V0

0

r0

r

FIGURE 4.15 The profile of the spherical repulsive potential.

PROBLEM 4.4.2 In Figure 4.15 we show the spherical repulsive potential: V(r ) = + V0

r ≤ r0 = 0 r > r0 .

Calculate the total cross section and the phase-shifts for elastic scattering of a lowenergy particle. Show that the total cross section is equal to Q0 = 4πr02 in the limit V0 (r ) → ∞. Considering that the classical hard sphere cross section is equal to Q0C = πr02 (see Exercise 4.3.1), explain why the quantum result is so different (Q0q = 4πr02 ). 4.4.2 Modified Effective Range Theory in Electron Scattering We have already established that the differential cross section in the elastic collision in basic quantum theory is expressed as σ (ε) = | f (θ ; ε)|2 , where the scattering amplitude f (θ) in the partial wave method is given by Equation 4.72 1 f (θ) = (2l + 1){(exp(2iηl ) − 1}Pl (cos θ ) 2ik l=0 in the partial wave method. Here k is the wave number of the incoming electron with energy ε. It follows that 4π Qm (ε) = 2 (2l + 1) sin2 (ηl − ηl+1 ) (4.76) k l and also that

Q0 (ε) =

4π (2l + 1) sin2 ηl . k2 l

(4.77)

These formulae are sometimes used to analyze the differential cross section of the electron and perform analytic extrapolations or integrations of the experimental differential cross section.

© 2006 by Taylor & Francis Group, LLC

72

Plasma Electronics: Applications in Microelectronic Device Fabrication

A method for making an analytic representation of cross sections for targets without permanent dipole moments was proposed by O’Malley and coworkers under the name of the modified effective range theory (MERT). This method yields analytic expansion of phase-shifts: πα 2 4α 2 tan η0 = −Ak 1 + k ln(a 0 ) − k + Dk 3 + F k 4 , (4.78) 3a 0 3a 0 tan η1 =

πα 2 k − A1 k 3 , 15a 0

(4.79)

where α and a 0 are the polarizability and Bohr radius, respectively. For all higher-order phase-shifts (l > 1), the Born approximation is sufficiently accurate: tan ηl =

π αk 2 . (2l + 3)(2l + 1)(2l − 1)

The parameters A (the scattering length), A1 , D, and F should be regarded as fitting parameters that may be obtained from transport coefficients of electrons and available cross section data. The value of parameter A may be determined using the dimension of the molecule at zero energy as Q0 = Qm = 4π A2 . For example, the best fit for argon is obtained for A = −1.459a 0 , A1 = 8.69a 03 , D = 68.93a 03 , and F = −97a 04 . MERT was found to work well for scattering of electrons on atoms at low energies where the elastic collisions are the only scattering process. In other words, this approximation is valid for electron–atom scattering below 1 eV. PROBLEM 4.4.3 Calculate Q0 , Qm for argon from 0 eV to 1 eV. Show that the Ramsauer–Townsend minimum (RTM) occurs when the contribution of both η0 and η1 are close to zero for the same energy. Show that the RTM occurs at different energies for Q0 and Qm and explain why.

4.5 Collisions between Electrons and Neutral Atoms/Molecules Collisions of electrons with gas phase molecules are very frequent and are the most important in a majority of low-temperature nonequilibrium plasmas. Accordingly, these collisions are described in greater detail than other processes. A specific property of the collision between an electron and a molecule is the large difference in mass, which is more than four orders of magnitude. Thus the recoil of the molecule in collisions is small and the energy loss of electrons in elastic collisions is very small, whereas the energy gain for molecules is negligible. This is the basic property that allows formation of nonequilibrium plasmas. On the other hand, the transfer of energy to

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

73

inelastic processes is close to 100% efficient. We can classify electron–molecule collisions by several criteria. The most important is division into elastic and inelastic processes. Another important distinction is between conservative and nonconservative processes of the electron, where, for example, ionization and attachment fall into the latter category. A theoretical description of the electron–molecule collision is quite complex and an analytic representation of the cross sections is generally impossible. Classical and semiclassical treatments are only of very limited use in specific processes. Quantum theory is therefore almost always necessary and the Hamiltonian consists of the part describing the isolated molecule, the kinetic energy of the electron, and the electron–nuclei and the electron– molecule interactions. Inasmuch as the electrons in the outer orbit in a molecule and the projectile electron are fermions, the Pauli principle dictates that the corresponding potential is repulsive and nonlocal. There is no classical analogue for this interaction. This effect is of importance when the electron is close to the target, that is, in the near region. There it will compete with numerous electrostatic interactions. If we move farther from the target the polarization effect becomes the dominant one. Polarization is induced by the projectile electron, and although the semiclassical picture is useful for understanding it, one needs quantum theory for practical calculations. At long distances r , the polarization potential is approximated by Vpol (r ) → −

1 α(r )e 2 , 8π ε0 r 4

(4.80)

where α is the polarizability of the molecule. If there is a permanent dipole moment or if the target is charged, then the interaction may be dominated in the far region by electrostatic terms. Molecules with a permanent dipole moment, although neutral, have a very long-range potential, and thus the cross sections may be large, especially for low-incident energies. As a result of these interactions and of the internal degrees of freedom of molecules, there are numerous electron–molecule collisions that may occur; these are listed in Table 4.1. 4.5.1 Resonant Scattering When an electron comes close to a molecule, it may be bound to the target molecule for a brief time (τ between 10−10 s and 10−15 s) depending on its initial energy and the structure of the molecule. When the time significantly exceeds the time required to cross the interaction region, we may talk about the formation of a compound state or temporary negative ion. The third term that may be used to describe this system is a resonance. Atoms in excited states may have positive electron affinity εa so that the energy of the bound electron is εex − εa . However, this system is unstable as compared with the ground state, so it may decay into a free electron with an energy close to εex − εa and the target molecule. In principle, if the state were long-lived it

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

74

TABLE 4.1

List of Electron–Molecule Collisions Process Elastic Electronic excitation Rotational excitation Vibrational excitation Dissociation Ionization Attachment Dissociative attachment Ion pair formation Superelastic collision

Reaction Scheme ε

e + M −→ e + M ε>εex

e + M −→ e + M j ε>εr

e + M(r ) −→ e + M(r ′ ) ε>εv

e + M(v) −→ e + M(v ′ ) ε>εd

e + AX −→ e + A + X ε>εi

e + M −→ e + e + M+ ε>εa

e + M −→ M− ε>εa

e + AX −→ A + X− ε>εi p

e + AX −→ e + A+ + X− ε e + M j −→ e + M

Energy Loss ∼ 2(m/M)ε εex

∆ε(Typical) ∼0

10 eV

εr

kT

εv

0.1 eV

εd

10 eV

εi

15 eV

εa εa εi p 0.01∼ –20eV

∼ kT

∼ eV

20 eV

could survive long enough to suffer an additional collision with another gas molecule and it would be stabilized, but usually the system is unstable and it decays by autodetachment. Resonances are characterized by their energy width ε, which may be associated with the compound’s lifetime through the uncertainty principle. PROBLEM 4.5.1 Calculate the energy width of the resonance that has a lifetime of 10−14 s. Describe the elastic and the inelastic resonant scattering on a molecule. There are two types of resonances: i. Feschbach resonances. Here a temporary potential well is induced when the target is excited by an electron and the electron is trapped. These lie from 0 eV to 0.5 eV below the parent excited state. ii. Shape resonances. The resonance state is above the parent state and the scattering process strongly depends on the shape of the potential well. It is evident that electrons must satisfy a very narrow energy condition in order to be able to excite the resonances. At the same time, enhanced interaction, due to a long-lasting bound state, will result in a much higher cross section, sometimes even by several orders of magnitude. It is difficult to distinguish whether scattering is resonant if the lifetime is very short (or the resonance is very broad). Yet resonant scattering is very useful in describing the shape of the cross sections. The term resonant usually refers to a specific situation where two processes have to be matched in a very narrow range of conditions (in our case energy) in order to interact. We use the term in this broader meaning elsewhere in this book. A specific term resonant scattering, as described in this section, refers to electron molecule collisions.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

75

4.6 Electron–Atom Collisions 4.6.1 Energy Levels of Atoms In the elastic collision between a light electron (mass m) and a heavier gas molecule (mass M), the loss of kinetic energy is on average equal to 2(m/M)ε, where ε is the kinetic energy of the electron. Thus for low energies, the efficiency of energy transfer in one elastic collision is 10−4 as compared with the value for typical inelastic processes. In order to understand inelastic collisions between an electron and atom, one should first consider the energy levels of an atom. The energies between different levels determine the energy of the transition ε = εk+1 − εk .

(4.81)

The threshold energy εex required for the upper excited state is equal to the excitation energy ε = εex ,

(4.82)

and below that energy the cross section for the electron-induced excitation is equal to zero. In Figure 4.16 we show the energy level for helium. Only the allowed radiative transitions for bound electrons are possible (both absorption and emission). The lifetimes of excited states that have allowed radiative transitions to lower levels are on the order of 10−9 s. Bound electrons can also be excited to triplet states, and these will eventually decay to the lowest excited triplet states that have no radiative transition to the ground state by dipole emission. Transitions by quadrupole and higherorder moments are weak, and the lifetimes of these levels are typically in the range of 102 s ∼ 100 s or even longer. These are the so-called metastable states. Of the lowest excitation levels of He (see Figure 4.16), two, 23 S and 21 S, are metastable states having lifetimes 7.9 × 103 s and 1.95 × 10−2 s, respectively. The levels such as 21 P and 23 P that have allowed transitions to the ground state are usually known as resonant levels, and their lifetimes are on the order of nanoseconds. If the electron collides with the excited metastable, then stepwise excitation or ionization is possible. This mode is more efficient, because the threshold for excitation of the higher level or the ionization is lower than that for the excitation from the ground state. On the other hand, the efficiency depends on the density of metastable atoms (or excited atoms in general). As the energy level increases, the energy gaps between excited states become smaller and smaller. The energy levels eventually converge to the ionization continuum where the excited electron is no longer bound to the atom forming an ion and free electron.

Exercise 4.6.1 Using the selection rules, explain the optically allowed and forbidden transitions in an atom.

© 2006 by Taylor & Francis Group, LLC

76

Plasma Electronics: Applications in Microelectronic Device Fabrication

25 24.6

Ionization 51D 41D

24

4922Å 6678Å

51S 41S

3965Å 5048Å

3 1P

31D

23 Energy (eV)

511P 4P

??

5015Å

+

He (2S) 53D 43D

53P 4 3P

4471Å

33D

31S

53S 4 3S

3188Å 4713Å

33P 3 3S

7281Å 5875Å

3889Å

7065Å

22

21

21P 23P

2.06µ

21S 20

1.08µ

He singlet 584Å

23S He triplet

19 0

He

11S

FIGURE 4.16 Energy levels of helium. Radiative transitions between singlet and triplet states are not allowed.

A typical term of an atom in a Russell–Saunders state is, for example, 43 P0 . The first number is the principal quantum number n (in this case n = 4). Values of orbital angular momentum L are denoted by capital letters: S(L = 0), P(L = 1), D(L = 2), . . ., (in this case L = 1). The spin angular momentum S is shown through multiplicity (2S + 1) and written as a superscript in front of the letter annotating the value of L (S = 1). The subscript to the right is equal to the total angular momentum J = L + S (J = 0), which has integer values for odd multiplicity and takes values 1/2, 3/2, . . . for even multiplicity. Parity is determined from the azimuthal quantum number for individual electrons li as (−1) li . The superscript to the right, if present, denotes parity (written for odd terms and not written for even terms). The magnetic quantum number is M. The selection rules are: i. J = 0, ±1, except for 0 → 0; ii. Parity must change; iii. L = 0, ±1, except for 0 → 0; iv. S = 0; v. l = ±1; and vi. M = 0, ±1.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

77

Rules (iii) to (v) are only approximate, but even strict rules may be broken if there is, for example, strong spin-orbit coupling in strong magnetic fields, due to nuclear perturbations or for autoionizing states. Rule (iv) does not allow transitions among different multiplicities (e.g., triplet to singlet transitions). Transitions with L= 0 are very rare. For our purposes, it suffices to note that transitions to the ground state of helium 11 S from 23 S and 21 S are forbidden because of the S = 0 rule in the former case and because of the L = 0 → 0 rule in the latter case. 4.6.2 Electron–Atom Scattering Cross Sections In Figure 4.17 we show a set of cross sections for electrons in helium. The elastic momentum transfer cross section at low energies is almost constant with a value close to 6 × 10−16 cm2 . At those energies helium is an excellent (and very rare) example of a constant cross section. Above 10 eV the elastic cross section falls. There are no inelastic losses until the incident energy exceeds 19.8 eV needed for the first electronic excitation. One should notice that the excitation to triplet states peaks close to the threshold and drops down rapidly due to the requirement of the spin exchange of the two bound electrons from s( 12 , − 21 ) to s( 21 , 12 ) by the aid of the incident electron. On the other hand, the cross section for excitation of singlet states is broad, peaking at around 100 eV, and resembles the ionization cross section in energy dependence. -15

10

Qm -16

Cross section (cm2)

10

Qi 1

3

Q2 P

Q2 S

-17

10

1

Q3 P

3

Q2 P Q others

-18

10

1

Q2 P 1

Q2 S 1 Q3 S

3

Q3 P -19

10

-2

10

-1

10

1

10

2

10

3

10

Electron energy (eV) FIGURE 4.17 Set of collision cross sections of electrons in helium as a function of electron energy. The subscript others denotes the summed effect of all other higher levels, and the subscript i indicates ionization. All other levels are denoted by their terms.

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

78

4p 4P7/2 35.25 eV

4p4

35 34 33 32

434.8 nm 4s 4P5/2 32.40 3V

4S4

Ar

15.76 3s2 3p5

2 2

15 Energy (eV)

+

P3/2 15.76 3V P1/2

3p5 3p6 3p7 3p8 3p9 3p10

5

3p 5p 14

419.8 nm

3p5 4p

13 420.1 nm

12

772.376 nm 750.4 nm 772.421 nm 811.5 nm

5

3p 4s 104.8 nm 106.7 nm

11 0

3s2 3p6

Ar

13.48 eV 13.33 eV 13.15 eV 13.08 eV

2p1 2p2 2p7 2p9 2p10

1s2 1s3 1s4 1s5

1

P1 P0 P1 3 P2 3 3

11.83 eV 11.72 eV 11.62 eV 11.55 eV

1

S0

FIGURE 4.18 Energy levels of argon. The Paschen notation is also given.

Quite often Paschen notation is used to label the levels, especially for heavier rare gases (see Figure 4.18). This is really just an assignment of levels in their order rather than a notation associated with quantum numbers. Thus, in this notation the ground state would be 1s1 , and the lowest metastable level would be 1s5 . Two resonant lines from the four low-lying levels are in the ultraviolet region, but most lines from the 4p and 5p states are in the visible and near-infrared regions and are often used for diagnostics of argon plasmas. Properties of the ground state, of the low-lying levels, and of the ionization limit for the rare gases are given in Table 4.2.

Exercise 4.6.2 In the case of argon (see Figure 4.18), the ground state is 1 S0 . Explain the allowed and forbidden transitions by using selection rules (and their possible breakdown). Discuss the selection rules in the case of other low-lying levels of the rare gases as given in Table 4.2.

© 2006 by Taylor & Francis Group, LLC

Elementary Processes in Gas Phase and on Surfaces

79

TABLE 4.2

Energies; Configurations; and Terms of Ground States, Low-Lying Levels, and Ionization Continuum for Rare Gases Gas

Configuration

Term

He

1s2

11 S0

Ne

[He]2s 2 2 p 6

2 1 S0

Ar

[Ne]3s 2 3 p 6

3 1 S0

Kr

[Ar]4s 2 3d 10 4 p 6

4 1 S0

Xe

[Kr]5s 2 4d 10 5 p 6

5 1 S0

Excited Resonant

Levels (eV)

Metastable

(eV)

Ionization

(eV)

2 3 P1 2 1 P1 3 3 P1 3 1 P1 4 3 P1 4 1 P1 5 3 P1 5 1 P1 6 3 P1 6 1 P1

20.96 21.12 16.67 16.85 11.62 11.83 10.03 10.64 8.48 9.54

2 3 S1 2 1 S0 3 3 P2 3 3 P0 4 3 P2 4 3 P0 5 3 P2 5 3 P0 6 3 P2 6 3 P0

19.82 20.61 16.62 16.72 11.55 11.75 9.92 10.56 8.32 9.45

2S 3/2

24.59

2P 3/2

21.56

2P 3/2

15.76

2P 3/2

14.00

2P 1/2

12.13

In the case of argon we have four low-lying levels: two of these are metastable (3 P2 and 3 P0 ), and the other two (3 P1 and 1 P1 ) are resonant (with allowed transitions to the ground state 1 S0 ). Transitions from the two metastables are strongly forbidden by the rule that only J = 0, ±1 transitions, except for 0 → 0, are allowed. In this case, the 3 P1 level has a resonant transition to the ground state in spite of the weaker rule that only S = 0 transitions are allowed. All rare gases with the exception of helium have a similar configuration of metastable and resonant states. Cross sections for electron–argon collisions are shown in Figure 4.19. One should observe that for energies below the threshold for the lowest metastable (11.55 eV in this case) there are no inelastic losses, as in helium. However, the elastic momentum transfer cross section has a broad minimum at around 0.23eV. This is the RTM, which is the first quantum effect observed for particles with nonzero mass. When the RTM is present, the minimum value of the cross section is two orders of magnitude lower than the maximum (and the value that one would expect without the RTM). RTM is present in heavier rare gases (Ar, Kr, Xe) and in many molecules. One should observe that above the four lowest levels there are a large number of excited levels, having transitions to those four levels including the two metastables. If one wants to calculate the excitation rate to a particular level, one needs to consider the excitation to all higher states and include the radiative cascading by properly taking into account the branching ratios from higher levels. In addition to excitation by electrons and radiativ ede-excitation, there are several oher processes that provide nonradiative transitions to lower or nearby resonant levels. These represent collisional quenching with other gas molecules, electrons, or walls of the chamber and are discussed later. We note, however, that some of the measurements of cross sections for excitation suffer from the effect of cascading. All higher levels are also excited if the

© 2006 by Taylor & Francis Group, LLC

80

Plasma Electronics: Applications in Microelectronic Device Fabrication -14

10

Qm Qi Cross section (cm2)

-16

10

Qex Q exm -18

Qex,(2p1)

10

Qex,(3p5) -20

10

4

Qi,4p( D) -2

10

-1

10

1

2

10

10

3

10

Electron energy (eV) FIGURE 4.19 Cross sections for electron–argon scattering as a function of electron energy. Excitation is represented by effective summed cross sections and is separated into excitation of metastables and excitation of higher levels. Three cross sections for specific transitions are added, two for excitations into a neutral and the other for ionization into an excited ion from the ground state neutral. These are employed for comparison between modeling and the optical diagnostics of a low-temperature plasma in Ar.

electron energy is sufficient. Short-lived radiative states decay and some of the transitions populate the excited state that is being studied. Thus the population of the excited state without collisional quenching will be Nj = Ndirect + j

k> j

Ak j

i 0). ( z)2

(8.20)

d is the ratio between the time step t and the characteristic diffusion time ( z)2 /D (i.e., the time it takes to move by z due to diffusion). c = vd

t (> 0) z

(8.21)

is the ratio between the time step and the characteristic drift time ( z/vd ), and c is also known as the Courant number. Finally, we rewrite Equation 8.19 as nim+1 = A nim ,

(8.22)

where A is a tridiagonal matrix. The distribution at tm+1 , nm+1 may be obtained by applying matrix A (m + 1) times starting from the initial assumed distribution n0 and bearing in mind that the values of the matrix elements

© 2006 by Taylor & Francis Group, LLC

238

Plasma Electronics: Applications in Microelectronic Device Fabrication

change both with time and with changing conditions in the spatial profile of the plasma. The difference between the densities obtained in subsequent iterations is:

2 ε = nim − nim−1 = nim − nim−1 , (8.23) i

and the difference ε may be associated with the error of calculations. If we consider eigenvalue λ of the matrix A, the solution for the number density may be written as nim = λm exp( jαi),

(8.24)

where j 2 = −1 and α is an arbitrary constant in wavenumber units. If λ is less than 1 then ε will disappear. Combining Equations 8.19 and 8.24 we obtain the solution for the eigenvalue λ: λ = 1 + 2d(cos α − 1) − jc sin α,

(8.25)

where the number density nim is stable under |λ|2 < 1. There are three possible situations (see Figure 8.4). i. Without diffusion (d = 0), λ is always greater than 1 for arbitrary values of c, and the solution is stable; ii. Without drift (c = 0), λ has a maximum for cos α = −1, and the solution is stable for d < 1/2; iii. With diffusion and drift, the system is stable for d< 1/2 and c < 2d, is: d < 1/2,

c < 2d (i.e., c < 1).

(8.26)

PROBLEM 8.2.1 Derive the relations in Equation 8.26 by reference to the trajectory of Equation 8.25 in the complex plane in Figure 8.4. By combining Equations 8.20 and 8.26 we obtain t

0 Φ(r, t) = 0

t

Φ(r, t) < 0

Φ(r, t+∆t) = 0

FIGURE 8.11 The moving boundary in material processing.

© 2006 by Taylor & Francis Group, LLC

t+∆t

Numerical Procedure of Modeling

253

surface evolution equation known as the Hamilton–Jacobi equation [8]: ∂(r, t) ∂ , (8.83) (r, t) = Rreact (r, t) ∂t ∂r where (r, t) is the surface function defined as

(r, t) > 0 in gas phase; = 0 on a boundary surface; and < 0 inside a material.

(8.84)

Note that the local surface reaction rate as the local surface velocity Rreact (r, t) is a negative value in etching and sputtering (see Equations 8.80 and 8.82) and a positive value in deposition (see Equation 8.81). An approach to the surface evolution based on the numerical solution of the Hamilton–Jacobi Equation 8.83 is named the Level Set method. The Level Set method was developed in [8] and is robust in two- and three-dimensional evolution problems.

References 1. Ferziger, J.H. and Penc, M. 1996. Computational Methods for Fluid Dynamics. Berlin: Springer Verlag. 2. Chua, O.L. and Lin, P.M. 1975. Computer-Aided Analysis of Electronic Circuit. Englewood Cliffs, NJ: Prentice-Hall. 3. Sharfetter, D.L. and Gummel, H.K. 1969. IEEE Trans. on Electron Devices ED-16:64. 4. Takewaki, H., Nishiguchi, A., and Yabe, T. 1985. J. Comput. Phys. 61:261–268. Nakamura, T. and Yabe, T. 1999. Comput. Phys. Commun. 120:125–154. 5. Ventzek, P.L.G., Hoekstra, R.T., and Kushner, M.J. 1994. J. Vac. Sci. Technol. B12:461. 6. Chantry, P.J. 1987. J. Appl. Phys. 62:1141. 7. Phelps, A.V. 1990. J. Res. Natl. Inst. Stand. Technol. 95:407. 8. Osher, S.J. and Sethian, J.A. 1988. J. Comput. Phys. 79:129. Sethian, J.A. and Strain, J. 1992. J. Comput. Phys. 98:231.

© 2006 by Taylor & Francis Group, LLC

9 Capacitively Coupled Plasma

9.1 Radio-Frequency Capacitive Coupling Plasma is generally sustained capacitively or inductively by a radiofrequency (rf) power supply. Part of the rf power input to the plasma is reflected back to the power supply as a reactive power. The maximum power dissipation in a plasma is supplied by an external rf source when the discharge plasma impedance Z is equal to the impedance at the external power source z (the maximum-power transfer theorem). Plasma impedance is determined self-consistently both by external plasma parameters and by the internal fundamental property of feed gas molecules. An impedance matching network is, therefore, required between the rf electrode and the power supply. A matching network with an equivalent capacitor, Cb (blocking capacitor), is typical. The discharge plasma connected by way of a capacitor to the rf power source is named capacitively coupled plasma (CCP). The value of Cb should be much greater than the sheath capacitance of the reactor.

9.2 Mechanism of Plasma Maintenance In this chapter, we consider a parallel plates reactor made by metallic electrodes as shown in Figure 9.1. A powered electrode is connected to an rf voltage source, Vrf (t), defined as Vrf (t) = V0 sin ωt,

(9.1)

through a blocking capacitor Cb. Here, V0 and ω are the amplitude and angular frequency, ω(= 2π f ), of the rf power supply, respectively. The other electrode as well as the side metallic wall is grounded to the earth. When an rf voltage Vrf (t) is applied to the parallel plate reactor in a vacuum, the current IT (t) between electrodes leads the applied rf voltage waveform Vrf (t) by π/2. The current is known as the displacement current or charging/discharging current between the two electrodes, and the external source has no power dissipation to the reactor; that is, V rf (t) · I T (t) = 0. 255 © 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

256

sheath

Cb bulk plasma V0 sinωt d

FIGURE 9.1 A typical CCP reactor.

When an rf voltage appropriate to sustain a discharge plasma is supplied between the two electrodes filled with feed gas molecules at pressure p, a finite discharge current flows through the closed circuit shown in Figure 9.1. Of course, the total current IT (t) is continuous as a function of axial position, whereas the magnitude of each of the elements of IT (t) is different in each position (see Chapter 7). When a low-temperature plasma is sustained between electrodes, the sustaining voltage at the powered electrode Vsus (t) is given by

Vsus (t) = Vrf (t) −

1 Cb

t

IT (t)dt,

(9.2)

−∞

where the second term on the right-hand side is the voltage drop between Cb to ensure a net direct current (DC) of zero in a periodic steady state. As a result, the sustaining voltage has a finite negative bias voltage expressed by the second term on the right-hand side of Equation 9.2. Capacitively coupled rf plasma is generally classified into two regions by the phase-shift between the total current IT (t) and the sustaining voltage Vsus (t). Low-frequency CCP is defined simply when the ions in the plasma satisfy the relation eff

Vd eff

1 > d, 2f

(9.3)

where Vd is the effective drift velocity of positive ions between parallel plates with distance d. That is, at low-frequency plasma, the ions can follow the local field change in time, and the current-sustaining voltage characteristics are resistive without phase difference. Note that the sheath behaves resistively at low-frequency CCP.

© 2006 by Taylor & Francis Group, LLC

Capacitively Coupled Plasma

257

1000

Minimum sustaining voltage (Vpp) [V]

SF6/N2(10%) 800 CH4(10%)/H2 600

H2

Ar

400 He

O2 200

0 10k

100k

1M

10M

100M

Frequency [Hz] FIGURE 9.2 Minimum sustaining voltage (experimental) in a quasi-symmetric parallel plate CCP as a function of applied frequency at 1 Torr and d = 2 cm.

On the other hand, at high-frequency plasma, it is difficult for massive ions to follow the instantaneous local field, eff

Vd

1 ≪ d. 2f

(9.4)

At high-frequency CCP, the total current IT (t) has a finite phase lead with respect to the sustaining voltage waveform Vsus (t), and the sheath behaves capacitively. Usually there exists a boundary between the low- and highfrequency plasmas at several MHz of the external frequency (see Figure 9.2), and the sustaining mechanism of the discharge plasma is completely different between the two regions.

Exercise 9.2.1 Discuss the maximum-power transfer theorem in the CCP system sustained by a sinusoidal voltage in Equation 9.1 in Figure 9.1. The circuit current I T (t) in the series circuit with plasma impedance Z and external variable impedance z is given by V rf (t)/(z + Z), and the average power dissipated in the discharge plasma, Pa v , is expressed by Pav =

1 T

© 2006 by Taylor & Francis Group, LLC

0

T

V sus (t) · I T (t)dt =

1 2 Re(Z) , V 2 0 |z + Z|2

(9.5)

Plasma Electronics: Applications in Microelectronic Device Fabrication

258

where T is the period of the external rf voltage source. The maximum power dissipation to the discharge plasma is obtained at the condition of Z = z (impedance matching). That is, the impedance of the discharge plasma Z must be matched to the conjugate of the power supply, z (maximum-power transfer theorem). 9.2.1 Low-Frequency Plasma As the amplitude of the applied voltage waveform is increased, the space between parallel plates filled by neutral gas will change from the Laplace field to Poisson’s field by way of the transition region from the Townsend discharge to glow discharge having a plasma phase (see [1, 2]). Low-frequency plasma is sustained by the electron multiplication of the secondary electrons ejected mainly by the impact of positive ions on the powered electrode by Auger potential ejection (see Section 4.11.2.4). The maximum flux and kinetic energy of ions incident on the powered electrode from the plasma occur at the phase of the lowest surface potential, that is, at 3π/2 in Equation 9.2. In order to keep a maximum current condition under a given rf voltage at the powered electrode, the space between the two electrodes is bifurcated into two regions, an ion sheaths region and a bulk plasma region. In front of the powered electrode, an ion sheath is constructed so as to allow a sufficient amount of electron ejection from the electrode and a sufficient acceleration of the secondary electrons, which generate ionization multiplications to maintain the low-frequency plasma. Therefore, a high field with a strong potential gradient is essential to the ion sheath (active sheath). A small ion sheath appears in front of the grounded electrode in the conventional parallel plates reactor. Note that, in an ideal symmetric electrode system, a pair of active ion sheaths is constructed in front of both electrodes during one rf period (see Figure 9.3). A bulk plasma region with quasi-neutrality (ne ∼ n p ) having a low (b)

2.8 0.0 20 D is ta 10 nc e (m m 0 )

W 10

V

5

120 0

-120

0

5 Time [µs]

I

10

0 -5

4 2 0 -2

Power [W]

240

Current [mA]

5.6 Voltage [V]

Λj (1011cm-3s-1)

(a)

10 0

5 (µs) im T e

-240

-10

-4

FIGURE 9.3 Typical example of the net excitation rate (a) and voltage-current characteristics (b) during one period in a low-frequency CCP at 100 kHz at 1 Torr in H2 .

© 2006 by Taylor & Francis Group, LLC

Capacitively Coupled Plasma

259

Ion flux

3 2

(a) (b)

1 0 0

200 400 600 800 1000 Energy (eV)

FIGURE 9.4 Time-averaged ion energy distribution incident on the powered electrode at a low-pressure rf plasma. The (a) low- and (b) high-frequency cases are shown.

field is formed between the two ion sheaths. The whole system is completely controlled by Poisson’s field. The total current in the sheath, which consists mainly of a conduction component of positive ions and electrons, coincides with the sustaining voltage waveform Vsus (t) without phase difference. Even in the case where the sustaining voltage is a sinusoidal waveform, the total current is nonsinusoidal in time, and the plasma potential behaves nonsinusoidally. Under the condition of a low-frequency CCP in which ions are influenced by the instantaneous sheath field, the energy of ions incident on the electrode through the sheath is characterized by a saddle-shaped profile, that is, bimodal distribution (see Figure 9.4). The energy dispersion between the maximum and minimum energy peaks decreases as the frequency of the rf source increases to the highfrequency region. The high degree of loss of ions from the bulk plasma to both electrodes or the side wall during a half-period of the low-frequency plasma exhibits some peculiar characteristics: i. Formation of low-density plasma with a strong and thick sheath; ii. Production of high-energy ions incident on the powered electrode; and iii. A considerably high value of the sustaining voltage. PROBLEM 9.2.1 A parallel plates reactor is connected to a low-frequency voltage source by a matching network consisting of inductance L m . Discuss the DC self-bias voltage at the powered electrode. The effective surface area of the powered electrode connected to an external rf source with Cb is usually different from that of the opposite electrode in a CCP. It is typical when the opposite metallic electrode is grounded to

© 2006 by Taylor & Francis Group, LLC

260

Plasma Electronics: Applications in Microelectronic Device Fabrication

the earth as well as the grounded chamber wall. Under these circumstances, the asymmetry of the discharge plasma is enhanced as there exists a large difference in the effective area between the two electrodes.

Exercise 9.2.2 Consider the pair of asymmetric parallel plate electrodes with the surface areas AP and AG shown in Figure 9.1. Estimate the ratio of each of the drops in sheath potential in front of both electrodes, Vsh P and VshG , in the case of a space-charge-limited regime at low frequency. In a space-charge-limited regime, the ion current density incident on the sheath with width of d is given by the Child–Langmuir expression, 3/2

Jp =

K o Vsh 1/2

Mp d 2

,

(9.6)

where K o is constant, and Mp is the mass of the ion. Provided that the ion current is dominant in the sheath, the ion currents incident on both electrodes during one period are equal to each other; that is, J p P AP = J pG AG . Assuming that the time-averaged potential drops in the sheaths with capacitances C P and C G are related to the total charge Q flowing to each of the sheaths as Q = C P Vsh P = C G VshG ,

(9.7)

where the two sheath capacitances are given by C P = ǫ0 AP /d P and C G = ǫ0 AG /dG , respectively. The ratio of the sheath voltage is derived from the above two equations as a function of the electrode area: Vsh P = VshG

AG AP

2

.

(9.8)

This relation is valid for low-frequency plasma or DC glow discharge under the condition that the main sheath current is supplied by the ion conduction. 9.2.2 High-Frequency Plasma As the frequency of the voltage supply increases, the ions will have a finite phase delay with respect to a time-varying instantaneous field. This means that the ion flux incident on the powered electrode and the number of the secondary electrons will decrease as the rf frequency satisfies the relation (9.4). At the same time, due to the difference of the phase between IT (t) and Vsur (t), the number of electrons and ions released to the electrodes will decrease. In turn, electrons released from the bulk plasma toward the powered electrode will be effectively reflected in the positive ion sheath as wave-riding electrons. The reflected electrons will be accelerated toward the bulk plasma through the sheath, and ionization multiplication will occur at the boundary between the sheath and bulk plasma (see Figure 9.5). This is a sustaining mechanism in a high-frequency plasma. As the external frequency increases, electrons

© 2006 by Taylor & Francis Group, LLC

Capacitively Coupled Plasma

261 (b)

2.1 0.0 20 D is ta 10 nc e (m m

0

0 Time (ns)

)

0

-60

74

0 -12

-30 37 (ns) im T e

37

I

1.2

24 12

30

74 0

W

-24

0.6 -0.6 0

Power [W]

V

60

Current [mA]

4.1 Voltage [V]

Λj (1011cm-3s-1)

(a)

-1.2

FIGURE 9.5 A typical example of the net excitation rate (a) and voltage-current characteristics (b) during one period of a high-frequency CCP at 13.56 MHz in H2 .

in addition to ions will gradually enter into their trap in space. Due to the presence of electron trapping inside the reactor, an apparent field of ambipolar diffusion will drop in front of the powered electrode in a time-averaged fashion. This will introduce a reduction of the minimum sustaining voltages as shown in Figure 9.2. The boundary between the low- and high-frequency plasma under conventional external conditions will appear at approximately several MHz. When a sinusoidal high-frequency voltage is supplied at the powered electrode, the total current in the sheath, consisting of the predominant displacement current and the conduction of electrons and ions, has a sinusoidal waveform. The capacitive sheath results in a sinusoidal variation of the plasma potential. The high-frequency plasma has several proper characteristics: i. Formation of high-density plasma with a weak thin sheath; ii. A relatively low sustaining voltage; iii. Production of low-energy ions incident on an electrode; and iv. A sinusoidal current waveform IT (t) leading the sustaining voltage waveform Vsus (t) is formed, and the discharge is defined as capacitive. The electron charge flowing into the blocking capacitance Cb through a small powered electrode during the positive potential should be equal to the positive ions during the rest of one period in a periodic steady state. The great difference between the mass of the electron and that of the ion will cause an excess negative charge in the capacitor during one period. Therefore, a negative bias voltage Vdc to the small electrode is needed in order to keep the zero net DC current in a periodic steady state through the Cb . That is, the surface of the powered electrode is negatively biased as Vsus (t) = Vrf (t) − Vdc (t).

© 2006 by Taylor & Francis Group, LLC

(9.9)

Plasma Electronics: Applications in Microelectronic Device Fabrication

262

TABLE 9.1

Classification of Sheaths Formed in Discharge Plasmas Sheath

Circumstances

Positive Ion

Position Formed

Physical Law

In Front Of

Passive (floating)

Without ionization

Wafer without bias Electric probe Reactor wall

Ambipolar diffusion

Active

Production of e/ion Dynamic sheath

rf powered electrode DC cathode Biased electrode

Plasma maintenance Plasma maintenance Energetic ion injection

Negative ion plasma

Instantaneous anode

e & n-ion acceleration

Negative Ion Active

Here, −Vdc (t) is the negative self-bias voltage with time variation. The timeaveraged DC self-bias voltage, Vdc = Vdc (t) =

1 2π

2π

Vsus (t) d(ωt),

(9.10)

0

is practically observed in the experiment. At high-frequency CCP, under which the ion transit time across the sheath is sufficiently longer than the half-period of the external source, the ion incident on the electrode reflects the time-averaged sheath characteristics (see Table 9.1). As a result, a single-peaked energy distribution is formed with a maximum at about the time-averaged sheath voltage (see Figure 9.4). The sheath formed in discharge plasmas is classified in Table 9.1. PROBLEM 9.2.2 At low pressure a collisionless sheath with thickness dsh , bulk electrons with average velocity < v > diffusing to the plasma sheath boundary will interact electrically with the moving boundary having an velocity V sh (t). This is not a binary collision between the electron and the neutral molecule but a wavelike interaction resulting from the long-range Coulomb interaction. Provided that the interaction is perfectly elastic under dsh /v > 2π/ω, the reflected electrons will change their velocity from − < v > → < v > + V sh ,

(9.11)

close at instantaneous cathode phase. Therefore, the phenomenon is a collisionless heating and is referred to as a stochastic heating of electrons, Fermi heating, or wave riding. Derive the relation 9.11 from the momentum balance of the electrons.

© 2006 by Taylor & Francis Group, LLC

Capacitively Coupled Plasma

263

9.2.3 Electronegative Plasma A conventional low-temperature rf plasma consists of electrons, positive ions, and feed gas molecules as the components. In a reactive plasma for etching, electronegative gases are widely used due to a strong chemical reactivity of the feed gas or the dissociated molecules (radicals) on a surface. There are hidden and interesting characteristics of an rf plasma for dry etching. That is, these reactive gases have the ability to produce negative ions by dissociative or nondissociative electron attachment with a threshold of several eV or thermal energy (see Section 4.8.2). With increasing pressure, the percentage of negative ions in the plasma increases due to the increase of the degree of spatial trapping of massive negative ions and of the collisional chance of electron attachment (see Table 9.2). The quasi-neutrality in the bulk plasma is realized as n p ∼ (ne + nn ).

(9.12)

The electronegativity Hen is defined by the terms of the number density of negative and positive ions, nn and n p , as Hen =

nn . np

(9.13)

Fully negative ion plasma is performed under Hen = 1, though the plasma is not maintained without electrons. In an electronegative plasma with densities ne , n p , nn , and N, the macroscopic measure of the plasma is characterized by both the electronegativity Hen and the degree of ionization Hdi given by Hdi =

np . N

(9.14)

Let us consider again the CCP system consisting of light electrons and massive positive ions as the components of the charged particle. The system TABLE 9.2

Negative Ions in Processing Plasmas Feed Gas

Negative Ion

SF6

SF−j (6

Cl2 BCl3 CF4 C4 F8 CF3 I O2

Cl− Cl− F− , CF−j F− , CF−j I− , CF−j O− , O− 2

SiH4 H2 /N2 HCl

SiH− H− Cl−

© 2006 by Taylor & Francis Group, LLC

≥ j ≥ 1),

Material Processing F−

Si, W etching Reactor cleaning Si, Al etching Si, Al etching SiO2 etching SiO2 etching SiO2 etching Oxidization, photoresist ashing Surface treaming a-Si:H deposition Reduction, low-k(organic) etching Si etching

Plasma Electronics: Applications in Microelectronic Device Fabrication

264

has typical plasma characteristics with a bulk plasma under quasi-neutrality (ne ∼ n p ) and with positive ion sheaths (i.e., electropositive plasma). That is, one is the presence of a positive space potential (plasma potential) Vs in the bulk plasma. The other is the strong sheath potential Vsh , which is essential to maintaining the plasma by the electron impact ionization. Both are caused by the great difference in speed between light electrons and massive positive ions. The general characteristics in the electropositive plasma will gradually change to a different phase in the presence of negative ions. Generally, in a high-frequency electronegative plasma consisting primarily of positive and negative ions, with fewer electrons, a thin, higher sheath field is realized as well as the presence of a strengthened bulk field. This is mainly caused by massive negative ions having a drift velocity of two orders of magnitude less than electrons. As a result, there exist three different regions of the plasma production during the half-period in electronegative plasmas at higher pressure, as shown in Figure 9.6a. One is the production by reflected electrons in front of the instantaneous cathode (I), another is by the bulk electrons in a high field (II), and the other is by electrons accelerated through a double layer close to the instantaneous anode (III). Also the phase difference between the total current IT (t) and the sustaining voltage Vsur (t) is shortened with increasing electronegativity Hen , as compared with electropositive plasma at all frequencies. In other words, the characteristics of an rf plasma change from capacitive to resistive in a strong electronegative plasma (see Figure 9.6). A typical electronegative plasma is formed in O2 , SF6 , Cl2 , CF4 , and so on with a finite collision cross section of electron attachment at high pressure. Electron capture by molecules excited to electronic or vibrational states is known to have a larger cross section as compared with that by the ground-state molecule. Negative ion production by way of electronically or vibrationally excited states will make a considerable contribution to the electronegativity in plasma, particularly in a high-density plasma. O2 (a 1 ) in oxygen plasma is an example of the excited states that strongly contribute to the electron

200 I

Voltage (V)

III

1.4

III II

0.0 20

II I 74

ce

an

10

m

(m

0 0

37 (ns) Time

1.4

W

0.7

100 0 -100 -200

0

Time (ns)

37

74 V

I

0 -0.7 -1.4

0.28 0.14 0 -0.14 -0.28

Power density (Wcm-2)

2.8

Current density (mAcm-2)

(b)

st Di

Λj[SF5+](1016cm-3s-1)

(a)

) FIGURE 9.6 Typical example of net excitation rate in electronegative high-frequency CCP in SF6 /N2 (10%) at 1 Torr.

© 2006 by Taylor & Francis Group, LLC

Capacitively Coupled Plasma

265 electrode total charge density

np >> (ne + nn) p-ion sheath np ~ (ne + nn) 0 bulk plasma np ω

FIGURE 10.10 Propagation of electromagnetic waves into a plasma with a spatial profile n0 z.

Exercise 10.3.2 When the intensity of electromagnetic waves is low enough not to perturb the bulk plasma, the plasma density is measured by using the principle that a wave propagating through a plasma has a phase shift relative to the wave propagating in a vacuum. Practically, a microwave is used in order to obtain a spatial resolution of ∼cm [3]. Derive a simple relation between the phase-shift φ and and the plasma density ne . The phase shift between two path φ(rad) is given as a function of the electron plasma frequency, ω pe (z) = (e 2 ne (z)/mǫ0 )1/2 : φ =

l

(k0 − kplasma )dl = k0

l

1/2 ω pe (z)2 dz. 1− 1− ω2

(10.22)

Then, φ = 2.82 × 10−17 λ0

ne (z)dz.

(10.23)

l

In particular, when the plasma is uniform in the radial direction of the ICP reactor, we obtain the plasma density ne (cm−3 ): ne = 1.18 f

φ , L

(10.24)

where f (Hz) is the microwave frequency and L(cm) is the effective width of the plasma investigated.

© 2006 by Taylor & Francis Group, LLC

288

Plasma Electronics: Applications in Microelectronic Device Fabrication

PROBLEM 10.3.1 The earth is surrounded with the ionospheric layer of plasma density, ∼106 cm−3 . A shortwave broadcasting (frequency: ∼MHz-30 MHz) is operated on the ground station. A satellite-based broadcasting is serviced at frequency greater than GHz. Discuss the difference of the available broadcasting-frequency.

References 1. Ventzek, P.L.G., Hoekstra, R.J., and Kushner, M.J. 1994. J. Vac. Sci Technol. B 12:461. 2. Miyoshi, Y., Petrovic, Z.Lj., and Makabe, T. 2002, 2005. IEEE Trans. on Plasma Sci. 30:130. Miyoshi, Y., Miyauchi, M., Oguni, A., and Makabe, T. IEEE Trans. on Plasma Sci. 33:362. 3. Auciello, O. and Flamm, D.L. 1989. Plasma Diagnostics, Vol. 1. San Diego: Academic Press.

© 2006 by Taylor & Francis Group, LLC

11 Magnetically Enhanced Plasma

Magnetron plasmas and electron cyclotron resonance (ECR) plasma are used in dry plasma processing at low pressure in which an external permanent magnetic field makes a significant contribution to the plasma maintenance by reducing electron losses to walls and electrodes. Helicon wave plasma and magnetic neutral loop discharge (NLD) are also magnetized plasmas. Collisionless electron heating through an electron cyclotron or a helicon wave is the main mechanism to sustain these plasmas at very low pressure. On the other hand, an induced magnetic field operates on an inductively coupled plasma (ICP) and surface wave plasma (SWP).

11.1 Direct-Current Magnetron Plasma Direct-current (DC) magnetron plasma has been widely used to deposit metallic film on a large area substrate in electronic and photonic device fabrication. Functional glass is prepared by material film coating. Sputter deposition of Cu atoms ejected from a copper target in a DC magnetron is practically used for interconnect in a trench or hole on SiO2 film. See Table 11.1. Typical DC magnetron plasma (planar magnetron) is maintained between two parallel plate electrodes, named the target and the substrate. Permanent magnets are arranged radially from the center on the back of the target (cathode) in order to supply a doughnutlike magnetic field in front of the target (see Figure 11.1). DC magnetron discharge is usually sustained at about 200 V in nonreactive Ar at several mTorr under a permanent magnetic field paralllel to the target, several hundred Gauss. At the low-pressure condition, an external magnetic field is essential for the maintenance of a plasma in a DC source, and the effective lifetime of electrons in the magnetron is prolonged (see Exercise 11.1.1). In particular, the net ionization rate is greatly enhanced at a point where the magnetic field component parallel to the electric field is zero, as compared with the plasma under no magnetic field. The Lamor radius of ions is on the order of centimeters in contrast to the sheath thickness with several mm in front of the target. Thus, ions in magnetron plasma are considered to be uninfluenced by the external magnetic field. Inert Ar is used as the feed gas in a magnetron sputtering plasma for several reasons. Inert Ar is chemically nonreactive on the target surface, the massiveness of the Ar+ ion enhances sputtering on the target, and Ar is highly 289 © 2006 by Taylor & Francis Group, LLC

290

Plasma Electronics: Applications in Microelectronic Device Fabrication TABLE 11.1

Magnetron Plasma for Sputtering Type Power source Magnet Feed gas Admixture Target material Film depo.

Wire depo.

DC Magnetron

rf Magnetron

DC Permanent Ar O 2 , N2 Metal Metallic Oxide metal Nitride metal Cu interconnect

rf (13.56 MHz) Permanent Ar O2 , N2 Dielectric, metal Dielectric

abundant in the earth. Oxide or nitride metal film is processed in a DC magnetron with a metal target in an admixture of a small amount of O2 or N2 with Ar. The process is known as a reactive sputter deposition.

Exercise 11.1.1 Both the electric E(=Ek) and the magnetic field B(=Bj) are applied in a gas at a number density of N. Derive the neutral density N estimated by apparent electron changes to

N 1+

eB m

2

1 2 Rm

1/2

(11.1)

.

The momentum conservation of a single electron with mass m and velocity v is d (mv) + mvRm = e(E + v × B), dt

(11.2)

where E = Ek, B = Bj, v = (vx i + v y j + vz k), and Rm is the total collision rate of the electron. We simply consider the case of Rm independent of v. Then, v is 1 Rm eE eB eE (11.3) v= − , 0, − . m m R2 + eB 2 m R2 + eB 2 m

m

m

m

The electron mean free path λe under B is as λe ∼

eE v = Rm mRm

1 eB 2 Rm 1 + m

1 2 Rm

1/2 .

(11.4)

Equation 11.4 means that the gas number density changes from N to the above expression 11.1. A decisive effect of a magnetic field on plasma production is expressed under the condition of the electron cyclotron frequency (Larmor frequency) ωce (= eB/m) ≫ Rm .

© 2006 by Taylor & Francis Group, LLC

Magnetically Enhanced Plasma

(a)

291

symmetric axis

metallic wall

substrate

feed gas

z

target

r N

N

S

DC

yoke

(b) 129

Bz (Gauss)

Br (Gauss)

40

-150

-611

-340

30

30

z( m m )

-241

0

N

0S

r (mm)

88

z(

m

m

)

0

0S

N

88

r (mm)

FIGURE 11.1 A typical DC magnetron plasma reactor describing a magnetic field line (a); radial and axial components of B(z, r ) (b).

Figure 11.2a shows a typical structure of a DC magnetron plasma in terms of the number density distribution of electrons and ions obtained by a hybrid model (see Section 7.4). The density distribution has a strong doughnutlike peak at position r (34 mm, 6 mm) with Bz = 0 in front of the target. This is caused by the presence of the peak net ionization rate at the position. It means that the radial nonuniformity of the plasma density (i.e., of the ion flux to the target) is the property intrinsic in a magnetron plasma, though it is capable of sustaining a plasma at lower pressure, several mTorr, as compared with a DC glow discharge or radio-frequency (rf) capacitively coupled plasma (CCP). Figure 11.2b shows the potential distribution in the reactor configuration in

© 2006 by Taylor & Francis Group, LLC

292

Plasma Electronics: Applications in Microelectronic Device Fabrication

ne, np (1010 cm-3)

(a) 3.5

ne np

1.7 0.0 30 z

(m

m

)

0

0S

88

N r (mm) ne

np

Potential (V)

(b) 5 -98 -200 30 z (m

m

)

0

0S

N

88

r (mm)

FIGURE 11.2 Plasma density (a) and potential (b) distributions in a DC magnetron in Ar at 5 mTorr driven at 200 V in the reactor in Figure 11.1.

Figure 11.1a. A thin sheath region with a strong potential difference appears in front of the target, and the potential in the bulk plasma (plasma potential) is very low when the plasma is surrounded by the metallic reactor wall grounded to the earth. In these potential distributions, electrons are mainly produced by the collisional ionization at the radially localized region with Bz ∼ 0 in the sheath edge, and these electrons diffuse to the bulk plasma in the very low electric and magnetic fields. On the other hand, ions produced locally as the pair of the ionization are strongly accelerated to the target with a beamlike energy, and these ions sputter the target material. It is easy to estimate a local erosion profile by ion sputtering from the local ion flux incident on a target. Magnetron plasma is usually operated in a region of current source. Removal of the external magnetic field, after the magnetron plasma is formed, introduces the discharge plasma with extinction.

© 2006 by Taylor & Francis Group, LLC

Magnetically Enhanced Plasma

293

PROBLEM 11.1.1 Discuss the reasons why maximum ionization efficiency is realized at the point where the magnetic field crosses at a right angle with the electric field in front of the target (cathode) in a magnetron discharge. PROBLEM 11.1.2 An azimuthal drift current in the plasma ring in front of the target is on the order of few amperes. Estimate the magnetic field generated by the loop current of electrons and compare the value with the external permanent magnetic field.

11.2 Unbalanced Magnetron Plasma Almost all magnetic field lines from the north pole of the magnet arranged behind the target terminate at the south pole through the gas phase in the reactor. The magnetron plasma with these magnetic field arrangements is called a balanced magnetron (BM). An unbalanced magnetron (UBM) has a proper magnetic field configuration in which a finite degree of the field lines from the outer magnetic pole diverge to the substrate, though the rest of the lines finish on the inner pole behind the target (see Figure 11.3b). Sufficient

FIGURE 11.3 Comparison between a balanced and unbalanced magnetron (a) and (b).

© 2006 by Taylor & Francis Group, LLC

294

Plasma Electronics: Applications in Microelectronic Device Fabrication

plasma density and a positive ion current on a metallic substrate even at a large distance from the target can be achieved in the UBM as compared with the BM.

11.3 Radio-Frequency Magnetron Plasma Magnetron plasma with a dielectric target is sustained only by an external rf source, which is distinguished from a DC magnetron plasma with a metal target. An rf power source at 13.56 MHz having a voltage waveform, Vrf (t) = V0 sin ωt,

(11.5)

is usually used through a blocking capacitor Cb . Therefore, the target surface is negatively biased and the surface potential takes the form 1 Vsus (t) = Vrf (t) − Vdc (t) where Vdc (t) = Cb

t

IT (t)dt.

(11.6)

−∞

An rf magnetron sustained at 13.56 MHz has a weak temporal change of the electron density close to the target except for the region trapped deeply by the magnetic field. The other difference from the DC magnetron is the presence of the phase to release the spatiotemporal electron trapping by electromagnetic fields, based on E(r, t) = 0 twice during one rf period. This allows a radially spreading distribution of the net ionization rate in front of the target (see Figure 11.4 ) as compared with that in a DC magnetron. The phenomena increase the efficiency both of the target utilization and of a much radially uniform deposition on the substrate. Note that the value of Cb is carefully arranged in the case of the dielectric target, especially for a ferrodielectric target with a high dielectric constant. Figure 11.4 shows the spatial profile of the electron density at each of the external voltage phases of ωt = 0, π/2, π , and 3π/2 in Ar at 5 mTorr. A local area close to the target (P B ) with Bz ∼ 0 is weakly modulated in time, because the electrons under a large component of Br are spatially trapped by the E(r, t)× B(r) fields in front of the target. As a result, the sustaining mechanism of the rf plasma is classified into two regions depending on the radial position. One is that of the conventional rf CCP, that is, the ionization multiplication in front of the target by the reflected electrons. The other is the ionization multiplication by the secondary electrons caused by the positive ion impact on the target. That is, rf magnetron plasma is maintained by mixed mechanisms between an rf CCP and a DC glow discharge. At low pressure such that rf CCP cannot be sustained, the first mechanism disappears. Figure 11.5 shows the ion flux incident on the Cu target (a) and the erosion profile (b) as a function of radial position [1].

© 2006 by Taylor & Francis Group, LLC

Magnetically Enhanced Plasma

295

ωt = π/2 ne [1010cm-3]

ne [1010cm-3]

ωt = 0 1.2 0.6 0.0

8.0

S 4.0 N r[cm]

8.0 0.0

1.2 0.6 0.0

8.0

S 4.0 N r[cm]

4.0 z[cm]

8.0 0.0

ωt = π

1.2

ne [1010cm-3]

ne [1010cm-3]

ωt = 3π/2

0.6 0.0

8.0

S 4.0 N r[cm]

8.0 0.0

4.0 z[cm]

1.2 0.6 0.0

8.0

S

4.0 z[cm]

4.0 N r[cm]

8.0 0.0

4.0 z[cm]

FIGURE 11.4 Typical electron density distribution in an rf magnetron sustained in Ar at 5 mTorr at 13.56 MHz and V0 = 400 V.

11.4 Magnetic Confinements of Plasmas When the operating pressure of a plasma drops to the order of mTorr, electrons and ions produced in a plasma rapidly diffuse and are lost to the reactor wall by the lack of binary collisions between the electron (or ion) and neutral feed gas molecule. The multipolar magnetic field arrangement on the reactor side wall is effective for the confinement of electrons inside the reactor, when the mean free path of fast electrons is equal to or larger than the chamber size. A magnetically confined reactor is arranged by an array of parmanent magnets with N-poles and S-poles alternately, positioned outside the reactor as shown in Figure 11.6a. The reactor with surface cusp magnetic fields is widely used in the application of a low-pressure plasma to material processings.

© 2006 by Taylor & Francis Group, LLC

296

Plasma Electronics: Applications in Microelectronic Device Fabrication (a)

Ions flux(10

15

-3 -1

cm s )

0 -1 -2 Dielectric target -3 Metallic target PB

-4

Target

-5 0

20

40 60 r (mm)

80

Depth (arb.)

(b) PB

0

-1 Target 0

20

40 60 r (mm)

80

FIGURE 11.5 Ion flux incident on Cu target (a) and erosion profile (b) in an rf magnetron plasma.

N

B

Er

S

N θp

FIGURE 11.6 Magnetically confined plasma reactor. Arrangement of multipolar magnets and net ionization rate.

© 2006 by Taylor & Francis Group, LLC

Magnetically Enhanced Plasma

297

The magnetic mirror effect near the cusp of two magnetic poles suppresses the electron diffusion in the direction normal to the magnetic field line in addition to the electron reflection in the wall sheath on the reactor. A set of straight magnets is arranged in the axial direction on the outer surface of the cylindrical reactor. The resulting line cusp magnetic field near the chamber wall confines electrons and ions. Such a field configuration has the properties of a magnetic mirror and has a very small effect on the confinement of slow electrons. That is, with increasing the number, the electron confinement becomes stronger, and the escape of electrons from the loss cone becomes significant [2]. The electron mean energy rises near the wall due to the presence of the high-energy electrons trapped by the multipolar magnetic field. The net ionization rate is expected to have a local peak. The local electron energy increases when the magnitude of the magnetic field increases due to the increase of the trapped high-energy electron. There is an optimum for the number of the magnetic poles. The magnetic field has a very small effect on the confinement of slow electrons. At pressure range where the electron mean free path is less than the reactor dimension, the magnetic field has little effect on the electron confinement by the trapping.

11.5 Magnetically Resonant Plasmas We consider an elctromagnetic wave, propagating parallel to the applied magnetic field B in a cold and uniform plasma. The wave has a frequency in that ion motion is negligible , ω ≫ ω pi . It is known that there are two modes, Rand L-modes, for electromagnetic waves traveling along the magnetic field. Each of the refractive indexes, ξ+ and ξ− , are given by [3–5] ω2pe k2c2 , (11.7) =1− ξ±2 ≡ 2 ω (ω ± ce )(ω ± cp ) where ce = eB/m (< 0) and cp = eB/M (> 0) are the cyclotron frequencies of electrons and ions, respectively. Equation 11.7 exhibits that the R- and L-modes have cut-off frequencies, ω+ and ω− (see Chapter 10), ω± =

1/2 cp + ce ( p − e )2 . ∓ ω2pe + 4 2

(11.8)

We see that the rotational direction of the R-wave corresponds to the direction of the cyclotron motion of electrons in the magnetic field. Note that the denominator of Equation 11.7 becomes zero when the frequency ω of the R-wave approaches |ce |. Then the electrons are continuously accelerated and result in the absorption of the energy of the electromagnetic wave. It is known as the ECR between the electron and the wave. It is noted that the L-wave has no resonances with electrons but with ions.

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

298 (a)

microwave electron-cyclotron wave guide wave quartz window

(b)

quartz tube rf power

antenna coil B coil

parmanent magnet

ECR layer B

plasma wafer

rf bias

wafer

plasma

rf bias

FIGURE 11.7 Representative for magnetically resonant plasma. ECR plasma source (a) and helicon source (b).

ECR plasma is excited by the resonance mechanism of the R-mode described above. A typical ECR plasma source is driven at microwave power with 2.45 GHz in low pressure of mTorr. It is the principle to synchronize the electron cyclotron frequency |ce | with the UHF (microwave) frequency 2.45 GHz at B of 875 G by adjusting the external coil current. In the resonance point, electrons efficiently get the external microwave energy and a diffusive high-density plasma is produced at very low pressure in a reactor as shown in Figure 11.7a. Microwave power through a wave guide is fed into the reactor through a quartz window. Wave heated plasmas are described in detail by Lieberman and Lichtenberg [6]. PROBLEM 11.5.1 Calculate the magnitude of the magnetic field at the ECR condition of electrons in a collisionless plasma supplied by the microwave power at 2.45 GHz.

References 1. Kuroiwa, S., Mine, T., Yagisawa, T., and Makabe, T. 2005. J. Vac. Sci. Technol. B, 23:2218. 2. Takekida, H. and Nanbu, K. 2004. J. Phys. D (37:1800).

© 2006 by Taylor & Francis Group, LLC

Magnetically Enhanced Plasma

299

3. Boyd, T.J.M. and Sanderson, J.J. 2003. The Physics of Plasmas. Cambridge: Cambridge University Press. 4. Sturrock, P.A. 1994. Plasma Physics. Cambridge: Cambridge University Press. 5. Nicholson, D.R. 1983. Introduction to Plasma Theory. New York: John Wiley & Sons. 6. Lieberman, M.A., and Lichtenberg, A.J. 2005. Principles of Plasma Discharges and Materials Processing (2nd Edition). Hoboken: John Wiley & Sons.

© 2006 by Taylor & Francis Group, LLC

12 Plasma Processing and Related Topics

12.1

Introduction

Plasma processes (i.e., sputtering, deposition, etching, surface treatment, etc.) require information about the surface reactions of active species and their probability, as well as information on gas-phase collision/reaction processes and their cross sections. Physical or chemical quantities describing the plasma surface interaction are given by an effective surface reaction probability consisting of the sticking coefficient and the yield for etching or deposition. Historically, the collision cross section of the electron or ion in the gas phase has been continuously accumulated theoretically and experimentally as a function of the impact energy, in addition to the study of the quantum characteristics of mono- and polyatomic molecules in the field of atomic and molecular physics. In the same way, we expect that the surface reaction process and the probability of neutral molecules and ions with a material surface will be rapidly elucidated and accumulated by using first-principle molecular dynamics and measurements.

12.2

Physical Sputtering

Metallic film deposition is usually performed by using a direct-current (DC) magnetron plasma with a metal target in pure Ar (see Figure 12.1). A relatively large number of data are available on ion sputtering in ion–surface interactions. Most of these data were collected for systems between metal targets and rare-gas ions with energy ranging from several hundred eV to several hundred keV under a clean surface [1, 2]. Very little is known about systems with low-energy ions with ǫ p < 100 eV. Reactive sputtering is a technique widely used for deposition of compound materials, for example, oxides (Al2 O3 , Ta2 O5 , etc.) and nitrides (Si3 N4 ). A sputtering system of a pure metal target in admixture with reactive gas (O2 or N2 ) and Ar holds the advantage over a system in a compound target in Ar from the viewpoints of the stoichiometry of the film and the power density and thermal conductivity of the target. Chemical sputtering is generally considered to be a multistep 301 © 2006 by Taylor & Francis Group, LLC

302

Plasma Electronics: Applications in Microelectronic Device Fabrication Substrate St deposition

+

M , εp sputtered

impact

Y(ε) Target

erosion

FIGURE 12.1 Physical sputtering system.

process finally leading to the formation of a volatile molecule that escapes into the gaseous phase. The sputtering efficiency of the system between an ion with incident energy ε and angle θ and a solid surface is described by the sputtering yield Y(ε, θ) as shown in Figure 12.2 as examples [3].

Exercise 12.2.1 A zero-dimensional model of a reactive sputtering process is successfully used for compound film deposition in a DC magnetron sputtering with a metallic target in admixture with reactive gas and Ar. Derive the governing equations of the system in which a tantalum (Ta) target is sputtered in an Ar/O2 mixture to deposit Ta2 O5 compound on a substrate [4]. The reaction on the surface in this system is divided into three regions: the target; substrate; and side wall with areas of AT , A S , and AW (see Figure 12.3a). A fraction (1 − X) of the target material sputtered by the Ar+ ions with flux Ŵ p deposits on the substrate, and X arrives at the side wall. T , S , and W give the degree of the compound (Ta2 O5 ) formation on the target, substrate, and reactor wall, respectively. The values of T and S are a practical indicator of the reactive sputtering from pure metallic target ( T = 0) resulting in deposition of a stoichiometric compound film at the substrate ( S = 1). st , ss , and sw are the sticking coefficients of O2 to the metallic part of the target, substrate, and wall, respectively. The sticking of the sputtered particles on the target is assumed to be unity. YTm and YTc are the sputtering yields of the metal and compound material at the target. The number ratio of the reactive atom (O) between the reactive gas (O2 ) and the compound molecule (Ta2 O5 ) is denoted by h r . At the target surface, the balance equation of the compound is ∂ T (t)AT = h r Ŵ O2 st (1 − T )AT − Ŵ p YTc T AT . ∂t

© 2006 by Taylor & Francis Group, LLC

(12.1)

Plasma Processing and Related Topics (a)

Incident ion energy (eV) 100 300 500

101

1 +

Ar / Cu 10-1

3.0 Y(250 eV; θ)

2.0

10-2

1.0

Y(100 eV; θ) 10-3

0

Incident ion energy (eV) 1.2

0

50

100

150

200

250

10 eV

1.0 Sputtering yield Sticking probability

0 90

30 60 Incident angle (deg.)

(b)

Y(ε; θ)

Sputtering yield Y(ε)

303

15 eV 0.8 25 eV

0.6 +

Cu / Cu 0.4

35 eV Y(ε)

0.2

50 eV 75 eV 0 0

20 40 60 80 Incident angle (deg.)

100 eV 100

FIGURE 12.2 Sputtering yield Y(ε) and Y(ε, θ ) of Ar+ -Cu target (a), and Y(ε) of Cu+ –Cu target and sticking coefficient St(θ) of Cu on Cu substrate (b).

At the substrate surface, the balance equation of the compound is described as ∂ S (t)AS = b m h r Ŵ O2 ss (1 − S )AS + (1 − S )b m (1 − X)Ŵ p YTc T AT ∂t − S (1 − X)Ŵ p YTm (1 − T )AT , (12.2)

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication Substrate : AS

(a)

(1-ΘS) SS

ΘW

ΘS

positive ion

SW

ΓO2 Γp

ST

YTm (1-ΘT)

(1-ΘW)

reactive gas O2

Wall : AW

304

YTc ΘT

Target : AT 1

Deposition rate (arb.)

(b)

0.5

0 0

1 2 Reactive gas flow (arb.)

FIGURE 12.3 The 0th-order reactive sputtering model (a), and the predicted deposition rate of a compound film as a function of partial pressure (b).

where b m is the number of metal atoms in the compound molecule. In the same way, for the side wall, the surface metal balance is described as ∂ W (t)AW = b m h r Ŵ O2 sw (1 − W )AW + b m (1 − W )XŴ p YTc T AT ∂t − W XŴ p YTm (1 − T )AT . (12.3) The total supply q of the reactive gas into the reactor is equal to the consumption on the surface and the quantity pumped out to the outside. That is, q = st Ŵ O2 (1− T )AT + sw Ŵ O2 (1− W )AW + ss Ŵ O2 (1− S )AS + pr Vpump , (12.4) where pr is the partial pressure of the reactive gas and related to Ŵ O2 = pr /(2π MkT)1/2 , and Vpump is the pumping speed of the gas. Here, the plasma parameters, incident ion flux Ŵ p , partial gas pressure pr , and pumping speed

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

305

ppump are given. Unknown parameters, X, T , S , and W can be solved in the steady state from Equations 12.1 to 12.4. The average deposition rate per unit area on a substrate R depo is given as R depo = Ŵ p

AT {YTc T + YTm (1 − T )}(1 − X). AS

(12.5)

The deposition rate R depo in the reactive sputtering has a range with multivalues as a function of reactive gas flow, as shown in Figure 12.3b. 12.2.1 Target Erosion Magnetron plasma is widely applied to thin film deposition. The target in a magnetron discharge, that is, cathode, is eroded by physical sputtering by high-energy ion impact under a low-pressure and highly localized plasma condition. The target erosion is described by the sputtering yield, Y(ǫ p ; target) =

atoms removed , incident ion

(12.6)

for the system between an incident ion with energy ǫ p and target material. The databases are widely available in the literature [1, 2]. The number of target atoms ejected by the sputtering of ions with velocity v incident on the target surface dS from the magnetron plasma is estimated by the velocity distribution of the incident ion, g p (ǫ p , θ), as Y(ǫ p ; target)vg p (ǫ p , θ, r)dθdǫ p dt dS, (12.7) ρ dSdl = n p θ

ǫp

where ρ is the atomic number density of the target material, dl is the erosion depth during a small time dt, and g p (ǫ p , θ, r) is normalized to unity as g p (ǫ p , θ, r)dθdǫ p = 1. (12.8) ǫp

The sputter rate Rsp (r) is obtained by n p (r) dl Y(ǫ p ; target)vg p (ǫ p , θ, r)dθdǫ p . = Rsp (r) = dt ρ θ ǫp

(12.9)

The erosion depth profile of the target is estimated numerically by the time development of the ion sputtering in a magnetron plasma. PROBLEM 12.2.1 Even in a low-pressure magnetron plasma, ions affect a target with a finite angle of incidence. Then, we must consider the angular dependence of the sputtering yield Y(ε, θ ). Revise Equation 12.9 of the sputter rate by considering Y(ε, θ).

© 2006 by Taylor & Francis Group, LLC

306

Plasma Electronics: Applications in Microelectronic Device Fabrication

12.2.2 Sputtered Particle Transport Next we consider the flux and energy of the atoms sputtered from the target. The ejected flux of the target atom is given at the surface rt by Y(ǫ p ; target)vg p (ǫ p , θ, r)dθ dǫ. (12.10) Γs (rt ) = n p θ

ǫp

On the other hand, the energy spectrum of the sputtered atom is estimated by the Thompson formula [5], f s (ǫ; ǫ p , rt ) = A

1 − {(Ut + ǫ)/γ ǫ p }1/2 , ǫ 2 (1 + Ut /ǫ)

(12.11)

where A is constant. ǫ p and ǫ are the kinetic energy of the incident ion and the sputtered atom, respectively. Ut is the binding energy of a target material. γ is the energy transfer factor in the elastic collision between the incident ion and the target atom with masses of Mp and Ms , respectively, and is given by γ =

4Mp Ms . (Mp + Ms )

(12.12)

PROBLEM 12.2.2 Discuss the reason that the angular distribution of the ejected neutral is usually approximated by the cos θ-law in the physical sputtering process. The Thompson formula is derived under the condition that the sputtered atoms come from a well-developed collision cascade in a material. The collision cascade is realized in a system between an incident heavy ion and light atom in the material. In a system between a light incident ion with low energy and a target consisting of a massive atom, however, the ion from the sheath will be easily backscattered by a massive target atom and will knock off an atom in the top layer of the target. Then, the sputtering is caused by a single knock-on mechanism. A modified formula has been proposed for the system between a light ion and massive target atom [6]: γ (1 − γ )ǫ p 2 ǫ ln f s (ǫ; ǫ p , rt ) = A , (12.13) (ǫ + Ut )α+1 ǫ + Ut where α = 3/5 for the H+ –Fe system. The angular distribution of the ejected atom is described by cosine law. We consider the arrival flux of the sputtered atom at the substrate rs . First, the flux without collision in gas phase is given by 0 Γs (rs ; ǫ) = Γs (rt )exp(−NQl)dS, (12.14) St

where l is the distance between rt and rs . Q is the collision cross section between the sputtered atom and feed gas molecule. Here we estimate the

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

307

transport of the relaxed component Nsrel by collision with feed gas molecules. The neutral transport is described by the diffusion equation ∂ rel N (r, t) = Ds ∇ 2 Nsrel (r, t) + rel (12.15) s (r, t), ∂t s where Ds is the diffusion coefficient of the sputtered atom in the feed gas molecule. The production rate of the relaxed component rel s is given under the assumption that the sputtered atom is randomized in energy and direction after one collision with feed gas molecules. Then, from the difference of the ejected flux distribution at small distance (l, l + dl), 0 rel s (r) = −∇ · Ŵs (r).

(12.16)

In a steady-state (∂/∂t = 0), the spatial transport of the sputtered atom under relaxation is obtained by Ds ∇ 2 Nsrel (r) − ∇ · Ŵs0 (r) = 0.

(12.17)

The arrival flux of the sputtered atom, collisionally relaxed in the gas phase, is given by Ŵsrel (rs ) = −Ds ∇ · Nsrel (rs ).

(12.18)

As a result, the total flux arriving at the substrate at rs is Ŵstotal (rs ) = Ŵs0 (rs ) + Ŵsrel (rs ).

(12.19)

Figure 12.4 shows the spatial density distribution of Ar+ ions and sputtered Cu neutrals in a DC magnetron sputtering system operated between z

1.0 × 109(cm-3)

3.0 × 108(cm-3)

1.5 × 109

np

1.0 × 1010 nCu

1.5 × 109

1.5 × 1010

3.0 × 109

r N

S

N

FIGURE 12.4 Relations of the distribution between the ion density n p (r) and sputtered atom Ns (r) in a DC magnetron plasma.

© 2006 by Taylor & Francis Group, LLC

308

Plasma Electronics: Applications in Microelectronic Device Fabrication

TABLE 12.1

Chemical Vapor Deposition of SiH4 (+H2 ) Deposited Film a-Si:H, µc-Si:H Poly-Si a-Si:H, µc-Si:H a-Si:H

Process

Radicals

Conditions

Ref.

Plasma process Thermal process Catalytic process Photo process

H, Si, SiH2 , SiH3 SiH2 , H2 , H, Si —

400 K, 1–103 Pa 900 K, 10–103 Pa 2000 K, 0.5–10 Pa Room temperature

[7] — [8] —

axisymmetric parallel plates at 5 mTorr in Ar sustained at Vsus of −200 V and maximum B of 300 G on a Cu target surface.

12.3

Plasma Chemical Vapor Deposition

Chemical vapor deposition (CVD) is a process for fabricating thin films or particles using chemical reactions of gas molecules. CVD is classified according to the various forms of the external energy source into plasma CVD, thermal CVD, photo CVD, and catalytic CVD. In these different processes, chemically active species (precursors) produced by dissociation of gas molecules have a strong influence on the deposition. To distinguish it from CVD, a deposition through a physical vacuum evaporation or sputtering is termed a physical vapor deposition (PVD). The quality of the film deposited is influenced by the properties of the precursor molecules, which depend on the process and the external condition. There are several different methods for depositing Si films (see Table 12.1): low-temperature plasma processing, thermal decomposition, photo decomposition, and catalytic decomposition on a hot tungsten filament. 12.3.1 Plasma CVD Capacitively coupled parallel plates plasma is widely used for hydrogenated amorphous silicon (a-Si:H) deposition. A good-quality a-Si:H film with a deposition area of up to a few m2 can be manufactured at a very low-power condition, 50 mWcm−2 , in SiH4 /H2 in parallel plate capacitively coupled plasma (CCP). Plasma damage to the film quality caused by an ion impact during deposition is prevented by a wafer arrangement on the grounded electrode and by using a high-frequency source at 13.56 MHz and more efficiently at very high frequency, 100 MHz. Recently, microcrystalline silicon (µc-Si:H) thin film has been developed into the basic materials of large-area solar cell and thin film transistor (TFT) for a large-area flat pannel display. Hydrogenated amorphous carbon (a -C:H) or diamondlike carbon (DLC) is the other application of the low-temperature plasma CVD. a -C:H film is hard

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

309

and wear resistant. The head and disk surfaces in magnetic disk drives are usually coated with a thin layer of DLC to minimize wear and corrosion. Thin films deposited by plasma polymerization (surface grafting) have a large variety of applications. The plasma has a high ability to modify the surface. These materials are widely used as biocompatible films with hydrophilicity and are suitable for various biomedical and pharmaceutical materials, for example, optical lenses, implants, and drug delivery devices. Plasma polymerization is usually performed in a glass reactor excited by helical coils driven at 13.56 MHz (i.e., inductively) in carrier gas Ar at several hundred mTorr with monomer vapor. Excitation of the carrier gas takes place in radiofrequency (rf) Ar plasma, generating active species (ions and radicals) in the gas phase. These active species interact with the substrate and generate reactive sites (mainly free radicals) on the wafer surface. The monomer vapors in the plasma chamber readily react with these radicals, yielding grafted surfaces.

Exercise 12.3.1 Diamond thin film grows on a diamond substrate in a thermal plasma in CH4 /H2 . The surface under deposition is divided into two sites: the diamond lattice growth with probability D and nth layers of amorphous carbon with probability Cn . Atomic H and C, dissociated from the feed gases CH4 /H2 , in thermal plasma diffuse to the substrate surface. The thin film growth is highly selective on the site and highly competitive between deposition and etching and is modeled by the birth and death process in the stochastic process [9–11]. That is, on the substrate surface, the evolution equation is given in the form of the 0th-order simultaneous rate equations dD = kc Ŵ H C1 − ks ŴC D + ketch Ŵ H C1 dt dC1 = −kc Ŵ H C1 + ks ŴC D − ketch Ŵ H C1 − ks ŴC C1 + ketch Ŵ H C2 dt dCn = ks ŴC Cn−1 − ketch Ŵ H Cn − ks ŴC Cn + ketch Ŵ H Cn+1 (n > 1), dt

(12.20) (12.21) (12.22)

where ks is the sticking coefficient of the C atom. ketch is the etching probability of a -C:H by an H atom. kc is the conversion probability of a -C:H to diamond lattice by an H atom, and D+

∞

Cn ≡ 1.

(12.23)

n=1

1. Derive the surface structure (C1 , Cn , and D) in the steady-state condition. 2. Derive the growth rate of the diamond film R depo .

© 2006 by Taylor & Francis Group, LLC

310

Plasma Electronics: Applications in Microelectronic Device Fabrication

By summing up Equations 12.20 to 12.22 in the steady state, we obtain the relation ∞ dD dCn + = −ks ŴC Cn + ketch Ŵ H Cn+1 = 0. dt dt n=1 Therefore,

Cn = χCn−1

where

χ=

ks ŴC . ketch Ŵ H

We obtain Cn as Cn =

(1 − χ n ) C1 . (1 − χ )

(12.24)

By substituting Equation 12.24 into Equation 12.23, we have ∞ (1 − χ n ) C1 D + lim C1 = 1 and D = 1 − . n→ (1 − χ ) (1 − χ ) n=1

(12.25)

Also, from Equation 12.20 we have C1 =

χD . c 1 + kketch

Equations 12.24 and 12.25 give D , C1 , and Cn as c (1 − χ ) 1 + kketch D= , c 1 + (1 − χ ) kketch C1 =

(1 − χ )χ , D 1 + (1 − χ ) kketch

Cn =

(1 − χ n )χ . C 1 + (1 − χ ) kketch

(12.26)

The deposition rate of diamond R depo is D R depo = kc Ŵ H C1 − ketch Ŵ H D, D where the etching rate coefficient ketch is less than the other rate constant, and

R depo ∼ kc Ŵ H C1 =

c ks (1 − χ ) kketch Ŵc

.

(12.27)

ks (1 − χ )ŴC ∼ kc ŴC . (1 − χ ) + kketch c

(12.28)

c 1 + (1 − χ ) kketch

At kc ≫ ketch , R depo (χ ) = At kc ≪ ketch , R depo (χ ) = ks (1 − χ )

© 2006 by Taylor & Francis Group, LLC

kc ŴC . ketch

(12.29)

Plasma Processing and Related Topics

311

12.3.2 Large-Area Deposition with High Rate High-quality thin film of hydrogenated amorphous silicon (a-Si:H) is fabricated under a low-energy ion impact in a very high frequency (VHF) plasma. A VHF source provides a higher density plasma with low-energy ions based on a high degree of spatial trap of electrons as compared with that at 13.56 MHz. However, inhomogeneity of plasma arises from the nonuniform surface potential on the powered electrode caused by a standing wave of the VHF voltage. Figure 9.9 shows the potential nonuniformity of the rod electrode as a function of position in a plasma of 2×109 cm−3 sustained at 100 MHz at 13 Pa in Ar. The numerical value, simulated by the transmission-line model (TLM) (see Section 7.5.3) reproduces the experimental observation. Numerical design of the surface potential uniformity is achieved by time-averaging the surface potential through the control of the modulation rate of the phase of the VHF source at the feed position in a one- or two-dimensional electrode system.

12.4

Plasma Etching

Dry etching for micro- and nanoelectronic device fabrication should be highly selective of one material over others and highly competitive with the deposition process. A competitive surface between deposition and etching is described by two physical quantities: the etching yield Yetch (ǫ) and the sticking coefficient St (θ) of incident active particles. Plasma etching has two extremely different phases, that is, isotropic chemical etching by neutral radicals and anisotropic etching assisted by energetic ion impacts (i.e., reactive ion etching). It is much more complicated in a practical etching that has a surface process between a purely physical sputtering and a spontaneous chemical etching as well as deposition as a function of radical-to-ion ratio incident on the wafer. The practical etching in ultra-large scale integrated (ULSI) manufacturing is mainly devoted to two processes: Si-gate etching and SiO2 contact hole (trench) etching (see Table 12.2 and Figure 12.5). In general, the atomic scale mechanisms that enhance the surface process by ion impact are i. Sticking coefficient of radicals; ii. Diffusion of etchant (i.e., surface coverage); iii. Impact damage; and iv. Volatile- and nonvolatile-molecule productions, and the like. Together, these effects mean that ion-assisted etching in a plasma process produces an increase in the local surface coverage of the etchant and enhances the effective etching yield compared with chemical etching under the same number of radicals incident on the surface. The etching yield Yetch (ε) is defined as the ratio between the number of removed substrate molecules and the number of the incident species on the

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

312

TABLE 12.2

Plasma Etching of Materials Materials

Feed Gas

Aluminum (Al)

Cl2

Nat. oxided-Al Silicon (Si)

BCl3 Cl2 , HBr, etc.

SiO2

C4 F8 , CHF3 , etc.

SiOCH (low-k) Organic low-k

C4 F8 /O2 N2 /H2 , NH3

Photoresist Comp. semicon. Metal (high b.p.) Transition group

O2 Comp. of halogen or H Compounds of F or Cl

Property Direct reaction between Al and Cl2 sidewall passivation of C compounds Deoxidization + ion-assisted etching Low-energy ion-assisted sidewall passivation of SiO2 High-energy ion-assisted, high-selectivity sidewall passivation of polymer (Ci F j ) High-energy ion assisted High-energy ion assisted sidewall passivation of Ci N j Ok Plasma trimming

Difficult for reactive etching

substrate. More specifically, the etching yield of ions is a function of the impact energy ε p and the incident angle θ , Y(ε p , θ). The etching rate R etch (r) of ions with velocity distribution g p (v, r) incident on the substrate r is related to the etching yield Y(ε p , θ ) when we define R etch (r) as n p (r) R etch (r) = Y(ε p , θ )vg p (v, r)dv/ g p (v, r)dv, (12.30) ρn where n p (r) is the number density of ions close to the wafer. ρn is the atomic number density of the substrate material. Reactive plasma

Cl2 - plasma

Cx Fy - plasma

resist

resist

Passivation layer (SiOClx)

Passivation layer (CFx-Polymer) Mixed layer

poly-Si

SiO2

(a)

(b)

FIGURE 12.5 Typical etching profile in Si-ULSI: Si-gate etching (a) and trench or hole etching of SiO2 (b).

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

313

PROBLEM 12.4.1 Reactive ion etching (RIE) is based on a first adsorption of radical species on a surface and successive ion impact with high energy. Discuss the necessary condition of radicals and ions incident on the surface in the RIE. 12.4.1 Wafer Bias The wafer to be etched is set on the substrate holder, which is usually electrically isolated from the reactor base potential. The simple relation between the wafer potential and the impact ion energy is briefly described in advance. 12.4.1.1 On Electrically Isolated Wafers (without Radio-Frequency Bias) When the wafer is set on an electrically insulated holder from the system, the wafer surface is kept at floating potential Vf l , which provides the ambipolar diffusion with eŴe = eŴ p on the surface in a time-averaged fashion. In a collisionless sheath, Bohm’s sheath criterion predicts the relation (see Section 3.6), Vplasma − Vf l =

kTe Mp ln , 2e 2.3m

(12.31)

where Vplasma and Te are the time-averaged potential and electron temperature in the bulk plasma. Mp and m are the mass of the positive ion and the electron, respectively. 12.4.1.2 On Wafers with Radio-Frequency Bias In the case of a high-energy ion-assisted etching, we must apply an rf bias voltage Vbias (t) on the holder. For sophisticated modern plasma etching, an ion-assisted etching with very high energy, 500 eV to 1 keV, is practical. Then, it is essential to perform the functional separation between plasma production and ion acceleration, because some degree of ionization is unavoidable in the active sheath in front of the biased wafer. As already described in Chapter 8, a negative DC self-bias voltage Vdc appears on the wafer under the ambipolar diffusion in the steady state. Accordingly, in a time-averaged fashion, the wafer surface is irradiated by positive ions with energy ε p , which is defined as ε p = e(Vplasma − Vdc ).

(12.32)

As a result, the impact energy of ions on the wafer is roughly estimated by the relation with/without bias voltage. Mp kTe ln : without bias (floating); ε p = 2 2.3m = e(Vplasma − Vdc ) : with rf bias; = e Vplasma : on the ground.

© 2006 by Taylor & Francis Group, LLC

Plasma Electronics: Applications in Microelectronic Device Fabrication

314

Another important particle on the surface exposed to the plasma etching is the chemically active radicals. The neutral radical transport is usually treated by the random motion in the gas phase. That is, the radical flux incident on a wafer with/without rf bias voltage is estimated by using the Maxwellian velocity distribution in the quasithermal condition Ŵradical =

Nr 4

8kTg π Mr

1/2

,

(12.33)

where Tg is the gas temperature. Nr and Mr are the number density and mass of the neutral radical, respectively. 12.4.2 Selection of Feed Gas Dry etching in a low-temperature plasma utilizes a physical and chemical reactivity on material surfaces based on ions and dissociated neutrals (radicals) in a plasma. Feed gases for plasma etching are carefully prepared under conditions (see Table 12.2): i. The reactive product caused by the surface reaction must be in gas phase (volatile) in order to remove the surface material, and thus the vapor pressure will be high for the reactive product (see Table 12.3); ii. The binding energy of reactive products must be lower than that of the material to be etched; and

TABLE 12.3

Reactive Product in Plasma Etching and the Boiling Point and Vapor Pressure Materials Al

Si

Fe

Ni W Co ∗ Sublimation

Reactive Product

Boiling Point (C)

AlF3 AlCl3 AlBr3 SiF4 SiCl4 SiBr4 FeF2 FeCl3 FeBr2 NiCl2 Ni(CO)4 -25 WF6 WCl6 CoF2 CoCl2

1291 183* 255* –90 57 5.4 ≤ 1000 319 684 1000 — 17* 346* 1200 1050

temperature.

© 2006 by Taylor & Francis Group, LLC

Vapor Pressure — 0.6 — ≤ 760 — — — — — — — — — — —

Plasma Processing and Related Topics

315

iii. It is preferable that the feed gas, the dissociated molecules, and the reactive products are all nontoxic and all have low global warming potentials. Admixture gases are usually used for plasma etching for the following purposes: i. Density control of the dissociated chemically active species (i.e., radicals); ii. Control of the absorption site on the material surface to be etched; iii. Control of the surface reactivity; iv. Improvement of heat transfer in gases and of surface cooling; and v. Control of the plasma density (impedance) by way of the plasma structure. The influence of each of the gases on surface etching is very complicated. Most of the feed gases used for plasma etching are greenhouse gases. That is, greenhouse gases absorb infrared radiation and trap energy in the atmosphere. The atmospheric lifetime and global warming potential (GWP) characterize the effect of the greenhouse gas. The GWP of a greenhouse gas is the ratio of global warming from one unit mass of a greenhouse gas to that of one unit mass of CO2 over a period of time. Hence this is a measure of the potential for global warming per unit mass relative to CO2 (see Table 12.4). 12.4.3 Si or Poly-Si Etching Si and poly-Si material are etched by halogen and halogen compounds. A fluorine atom can be used to etch the material without the assistance of ion impact, and isotropic etching is naturally realized. F atoms incident on the clean Si surface saturate the dangling bond and insert into the Si-Si bond, resulting in an ejection of volatile particles, SiF4 , as the major etching product

TABLE 12.4

Global Warming Potential (GWP) and Lifetime of Gases for Plasma Etching Feed Gas

GWP (Units of 500 yr)

Lifetime (yr)

CO2 CF4 CHF3 C3 F8 SF6 c-C4 F8 l-C4 F8 C3 F6 l-C4 F6

1.0 8.9 × 103 1 × 104 1.2 × 104 3.2 × 104 9100 100 5 × 10 5 × 10

1.8 × 102 5 × 104 2.6 × 102 2.5 × 103 3.2 × 103 3200 1.0 1.0 1.0

© 2006 by Taylor & Francis Group, LLC

316

Plasma Electronics: Applications in Microelectronic Device Fabrication

at room temperature: 4F + Si → SiF4 .

(12.34)

The chemical etching probability (yield) of F atoms and Si crystals is sensitive to the surface condition, contamination, roughness, and so on, and the value distributes between 0.025 and 0.064. With increasing substrate temperature (T S > 500 K), the fraction of SiF4 gradually decreases and that of SiF2 increases rapidly. On the other hand, the etching probability (yield) of crystalline Si is 0.005 for a Cl atom at room temperature, and the dominant etching product is SiCl4 . With increasing surface temperature, the dominant etching product becomes SiCl2 : 4Cl + Si → SiCl4 at TS < 500 K, 2Cl + Si → SiCl2 at TS > 750 K.

(12.35)

Ion energy-assisted etching of Si-crystal or poly-Si enhances anisotropic etching in an rf plasma in chlorine or bromine compounds. Therefore, the anisotropic etching of Si or poly-Si is performed on the wafer biased at less than 100 V in a high-density chlorine plasma in inductively coupled plasma (ICP) or rf-magnetron, and so on. It should be noted that the etching probability of crystalline Si by molecular F2 and Cl2 will be very small. PROBLEM 12.4.2 The volume density of neutral etching products is simply estimated without redeposition in a plasma as Netch =

R etch ρ Seff 1 , V R j ne + kpump

(12.36)

where Seff is the effective area of a substrate exposed to etching, V is the plasma volume, R j is the rate of the destruction processes of the species in gas phase, and kpump is the pumping speed. Derive the expression 12.36. Plasma etching is used for fine-pattern transfer in ULSI processing. The target of the present technology is the fabrication of sub-45 nm rule or less. The validity of the model depends directly on knowledge of both the gasphase collision processes and the physics/chemistry of the surface exposed to the plasma. Figure 12.6 shows the etching yield Y(ε) of Si as a function of incident ion energy [12]. Note here that in a typical plasma etching, the Cl flux is larger than the ion flux by a factor of 100. Attention should be paid to the difference of the yield between the nonreactive Ar+ ion and reactive Cl+ 2 ion. As described in Equation 12.12, the magnitude of the physical etching is the direct result of the mass ratio, Mp /Ms . Judging from the fact that the mass + ratio of Cl+ 2 in Cl2 plasma is similar to that of Ar in Ar plasma a predominant element comes from the chemical reactivity, that is, chemical etching, of the neutral radical species, Cl. In fact, this is confirmed from the values of the yield + of Cl+ 2 /Cl2 as compared with that in Ar /Ar in Figure 12.6. Active fluxes of

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

317

(a)

Yield (Si atoms/ion)

6 4

+

Cl2 /Cl2

2

1

Ar+/Cl2

8 6 4

Ar+/ Ar

2

0.1

8 6 4

0

500 Incident ion energy (eV)

1000

(b) +

Etching Yield (SiO2 /ion)

2.0

CF3

1.5

+

CF2

1.0

CF

+

Ar+

0.5

F+ 0.0 0

500

1000 1500 Ion energy (eV)

2000

FIGURE 12.6 Etching yield of Si (a) and SiO2 (b) as a function of ions from plasmas.

ions and neutral radicals incident on the surface and passive ejected flux of neutral etch products can be, in principle, extracted from a self-consistent modeling of the total system of the dry etching. However, it may be difficult to perform the modeling due to the lack of an available database. 12.4.4 Al Etching Al etching is generally performed by chlorine and the compounds. However, fluorine compound is not valid for this purpose, because the reactive etching product has a high boiling point and is usually nonvolatile (see Table 12.3). The surface etching is described by the direct reaction between Cl2 and Al: 3Cl2 + 2Al → 2AlCl3 .

© 2006 by Taylor & Francis Group, LLC

(12.37)

318

Plasma Electronics: Applications in Microelectronic Device Fabrication

As a result, the etching speed depends on the gas number density of Cl2 . On the other hand, in BCl3 the plasma density controls the etching speed because the surface reaction proceeds by way of the dissociated Cl2 in BCl3 plasma: e(plasma) + BCl3 → (Cl, Cl2 , BCl, BCl2 ) + e 3Cl2 + 2Al → 2AlCl3 .

(12.38)

That is, the etching speed in rf plasma in Cl2 is independent of the dissipated power, and the speed is proportional to the power in BCl3 plasma. This means that ion bombardment from the plasma has no effect on the etching rate, and isotropic etching is performed. The native surface oxide is removed by the ion impact in BCl3 plasma. 12.4.5 SiO2 Etching SiO2 etching for electrical contact holes or trenches in ULSI circuits is usually performed by a CCP. This process requires a high-energy ion-assisted etching with several hundred eV or 1 keV in order to maintain a high etching rate and high selectivity of SiO2 to the Si surface. For this purpose, a two frequency CCP with a different frequency source, functionally separated for sustaining a high-density plasma and for biasing the wafer, at each of two parallel electrodes has been adopted by a combination of VHF and low-frequency sources. SiO2 thin film is generally etched by an rf plasma in a fluorocarbon C j Fk gas system. In SiO2 the characteristics of a fluorocarbon polymer deposition as well as the high-energy ion-assisted etching determine the feature profile of the hole or trench. The control of the profile with a high aspect ratio is performed by the sidewall passivation film of the polymer. The selective etching of SiO2 over Si and photoresist is due to the selective formation of protective fluorocarbon polymer film over the Si surface. In a steady plasma etching, a mixed amorphous interfacial layer, Sil Cm Fn , is formed on SiO2 under impact of energetic CF+ 3 ions. In actual practice, the apparent etching yield of SiO2 in the continuous plasma irradiation is given for the value of a mixed interfacial layer rather than for the pure SiO2 surface. Carbon in the form of CFi radicals and CF+j ions from fluorocarbon plasma reacts with the oxygen on the SiO2 surface under energetic ion impact to form volatile products, whereas fluorocarbon is deposited on the sidewall to form a protective layer of Ci F j polymers that inhibit lateral etching. This leads to “anisotropic etching.” Low-k materials with a relative permittivity ǫr smaller than 3.8 of SiO2 are used as the dielectric in a multilayer interconnect system in ULSI. CCP maintained in admixture of H2 and N2 is used to etch organic low-k materials. The etching yield is shown in Figure 12.7. PROBLEM 12.4.3 Discuss the effects of additive gases (Ar, O2 , CO, H2 ) on SiO2 etching in fluorocarbon plasma (see Table 12.5).

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

319

2.0 N2 /H2 plasma

Sputtering yield (C / ion)

1.5 N2 plasma

1.0

Ar plasma 0.5 H2 plasma 0.0

0

400

800

1200

Ion energy (eV) FIGURE 12.7 Etching yield of organic low-k as a function of ions from plasmas.

12.4.6 Feature Profile Evolution The space and time evolution of a patterned wafer surface exposed to particles of positive ions and neutral radicals from a plasma as shown in terms of the velocity distribution of ions in Figure 12.8 is estimated by the Level Set method based on the Hamilton–Jacobi-type equation under a moving boundary [13] (see Chapter 8): ∂ (x, z, t) = R etch (x, z, t)|∇|. ∂t

(12.39)

Here, R etch is the etching rate (speed function) of a material surface. The surface as a function of Cartesian coordinates (x, z) and time t in Figure 12.9 is TABLE 12.5

Function of Additive Gas in SiO2 Etching in Fluorocarbon Plasma Gas O2 H2 CO Ar

Property Decelerates polymerization through the production of volatile CO, CO2 , COF2 Scavenges [F] to form volatile HF, increasing the [C]/[F] ratio, and shifting the chemistry from an etching to a polymerization Tunes gas to increase the selectivity in SiO2 /Si Increase of relative CF Serves as a buffer gas for control of the dissociation of C j Fk

© 2006 by Taylor & Francis Group, LLC

320

Plasma Electronics: Applications in Microelectronic Device Fabrication

p

1.0

0.0 0

+20 0

1500 -20 En

erg

y(

eV

Angle

)

(deg)

FIGURE 12.8 Example of the energy and angular distributions of ions incident on a wafer as a function of radial position.

z ion trajectory Φ=0

(i+1, j) Φ0

r0 n rp (i+1, j+1)

m

(1-s)

Gas

(i, j+1)

Solid

x

FIGURE 12.9 Schematic diagram of the Level Set method and the surface evolution.

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

321

defined as (x, z, t) = 0.

(12.40)

That is, the gas and solid phase are given at > 0 and < 0 at t, respectively. The dielectric surface (r, t) = 0 is usually locally charged by ions and electrons incident on the surface during etching, and the metallic surface is kept at equipotential. A brief description of the Level Set method is given below. First, we investigate the relation between the time constant τ0 for one monolayer etching and the time step to trace an ion trajectory ttr in order to estimate a surface evolution t under the following physical requirement: ttr ≪ τ0 < t. Also t is adjusted to satisfy the condition that a renewal of the surface during t does not exceed the spatial mesh size, x and z; that is, R etch t ≪ x(and z). Step 1: At a position rintl unaffected by a topographically and locally charged wafer surface, the velocity distribution g p(e) (v, rintl ) and the density n p(e) (rintl ) of positive ions and electrons are given from the plasma structure in the reactor simulated, in advance, as the initial conditions of ions and electrons incident on the wafer surface (see Figure 12.9). Then, the ion (and electron) trajectory incident in a trench from rintl is traced by Monte Carlo simulation with time step ttr by using Poisson’s equation and Newton’s equation under the consideration of collisions with the gas molecule and on the surface. At each position r p (ttr ) of the ion trajectory, (r p ) is interpolated from i, j at four grid nodes surrounding r p as (see Figure 12.9) p (r p ) = (1 − r )(1 − s)i, j + (1 − r )si, j+1 + r (1 − s)i+1, j + r si+1, j+1 ,

(12.41)

where r = (x p − xi )/(xi+1 − xi ) and s = (z p − zi )/(zi+1 − zi ). When p (r p ) > 0 is satisfied, the trace of the ion flight is continued at ttr + ttr . Otherwise, if p (r p ) < 0 is satisfied, go to Step 2. Step 2: We estimate the point r0 that intersects with the surface, (r0 ) = 0, from (r p ) and the value one step before (r p−1 ) at ttr − ttr as p x p−1 − p−1 x p p yp−1 − p−1 yp , . (12.42) r0 (x0 , y0 ) = p − p−1 p − p−1 When the ion with energy ε p is reflected at the material surface having a reflection coefficient αref (ε p ), that is, when the condition ξ ≤ αref (ε p ) is satisfied, the trace of the ion trajectory in gas phase is continued at Step 1. Here, ξ is a uniform random number distributed over [0, 1].

© 2006 by Taylor & Francis Group, LLC

322

Plasma Electronics: Applications in Microelectronic Device Fabrication

In the case of ξ > αref (ε p ), the information of the flux velocity Γ(v0 , r0 ) at the surface is transcribed into the adjacent two-grid nodes (see Figure 12.9):

Γ(v)i+1, j = 1 +

Γ(v)i, j+1 = 1 +

s 2 + (1 − r )2 r 2 + (1 − s)2

1/2 −1

r 2 + (1 − s)2 s 2 + (1 − r )2

Γ(v0 , r0 ),

1/2 −1

Γ(v0 , r0 ).

(12.43)

Then, the incident angle of the transcribed flux vector θ with respect to the normal vector n = ∇/|∇| at both nodes is obtained by (r) and the derivative of the nodes: ∂ · vx + ∂ · vy ∂x ∂y −1 . (12.44) θ = cos

∇φ · v

Then we have the component of the etching yield at the node as a function of the ion energy ε p and incident angle θ as Y(ε p , θ). The above procedure is carried out for the number of ions at initial position rintl by considering g(v, rinitl ) and n p (rintl ). Then, the flux velocity Γ(v, r) at each node adjacent to the surface = 0 is accumulated during t. Step 3: After tracing the ion trajectories incident on a trench structure from a plasma during t = m ttr (m ∼ 100), the accumulated flux velocity Ŵ(ε p , r), energy ε p , and angle θ at each of the grid nodes adjacent to the surface = 0 enables us to estimate the new surface = 0. That is, at each of the nodes, the etching rate (speed function) is given as R etch (r) = n pintl vg p (v, r)Y(ε p , θ )dε p dθ/ vg p (v, r)dε p = Γ(ε p , r)Y(ε p , θ)d ε p d θ. (12.45) As a result, the new value of (r, t) is obtained at each node from Equation 12.39 t+t R etch (r, t)|∇(r, t)|dt. (12.46) (r, t + t) = t

Step 4: By using the information of the renewed surface ((r, t) = 0), at each grid surrounding the surface (r, t) is redefined. At the same time, the local charge distribution on the new surface is also renewed by using the previous ones. Successive simulation of the etching is performed by repeating the loop from Step 1 to Step 4. The same procedure is adopted for chemical etching by neutral radicals incident on the surface. In general, the neutral radicals have a Maxwellian velocity distribution with a temperature of Tg . Plasma etching is a more-orless competitive process between deposition and etching. As described in Section 12.3, neutral radicals are the predominant species for deposition, and

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

323

FIGURE 12.10 A typical example of feature profile evolution by plasma etching: (a) without charging and deposition, (b) with charging and without deposition, and (c) with charging and deposition.

etching is achieved by positive ion impact in addition to the chemical etching of neutral radicals. In particular, the surface reaction of neutral species is a function of surface temperature as predicted by Arrhenius’ equation. It is possible to carry out the simulation of the competitive process by considering a surface evolution by neutral radicals and ions using the Level Set method under the database of Y(ε p , θ ) of ions and St (θ) of radicals. Figure 12.10 exhibits self-consistent results of a feature profile evolution based on plasma reactor-scale and feature-scale simulation. PROBLEM 12.4.4 Discuss the string model of the feature profile evolution in etching in comparison with the Level Set method. 12.4.7 Plasma Bosch Process Plasma etching of a large hole/trench with high aspect ratio (depth-to-width) in Si is a basic process to fabricate integrated components for microelectromechanical systems (MEMS) [14]. Holes or trenches in MEMS range from 1 µm to 100 µm in width and from 10 µm to several 100 µm in aspect

© 2006 by Taylor & Francis Group, LLC

324

Plasma Electronics: Applications in Microelectronic Device Fabrication

ratio. The etching needs a high etch rate with anisotropy and selectivity to the mask material. Fluorocarbon plasma, maintained in CF4 , C4 F8 , SF6 , and so on, in admixture with O2 are used for the processes under the basic reactions that F isotropically etches Si wafer with rapid chemical etching, and a fluorinated silicon oxide passivating layer is formed on the sidewall to keep the anisotropic etching. A series of alternating processes of etching and depositing is effective in forming MEMS construction. The time-multiplexed deep etching is known as the Bosch process. One of the effective and rapid plasma processes consists of three steps: isotropic etching of Si/polymer in SF6 , polymer passivation in C4 F8 , and polymer depassivation in O2 . The resulting feature profile keeps a high anisotropy, though the etching is chemically isotropic. The Plasma Bosch process is widely carried out by using a high-density ICP reactor at pressure between 1.33 Pa (10 mTorr) and 13.3 Pa (100 mTorr). The dimension of the MEMS structure is usually comparable with the sheath thickness or larger. It means that the bulk plasma and sheath structure exposed to the substrate will change in time (plasma molding).

Exercise 12.4.1 In a plasma for deposition or etching, a large size polyatomic molecule may grow up in circumstances of high degree of dissociation of the feed gas. The generated species is a named particle, dust, or powder [15]. Discuss the tool for detection of these large polyatomic molecules in a plasma. The plasma potential is always a positive value during the cw operation. Therefore, positive ions and neutrals are detected through an orifice on the electrode or reactor wall. However, it is not easy to detect negative ions from the electrode. Technically, a pulsed operation of the plasma is introduced to measure negative ions in the afterglow phase. Mie scattering is an in situ, active procedure (see Table 12.6). 12.4.8 Charging Damage We have basically two types of materials to be etched in ULSI. One is metal or poly-Si with adequate electric conductivity. The other is dielectric SiO2 or low-k materials. These are known as Si-gate etching and the contact hole (via or trench) etching in SiO2 , which is affected by an anomalous etching caused by a local charging of the surface exposed to a plasma as shown in Figure 12.11. The surface resistivity changes with the flux composition from the plasma, that is, ions, electrons, and neutral radical species. TABLE 12.6

Detection of Polyatomic Molecule or Cluster in Plasmas Method Quadrupole mass spectrometer Time-of-flight mass spectrometer Laser Mie scattering

© 2006 by Taylor & Francis Group, LLC

Molecular Size 103

M/e < M/e < 105 d0 > µm

Mass Resolution

Ref.

— 5×103 —

[16, 17] [18] [19]

(b) Sheath Sheath

f e (ε,θ) f p (ε,θ)

f e (ε,θ) f p (ε,θ) E

E M+

e

Wafer surface

e

Wafer surface

Open area

Resist

Resist M+

SiO2

Poly-Si

Tunneling current

SiO 2

Metal

Damage current

Si

Plasma Processing and Related Topics

(a)

SiO2 Damage current

Si Cb Cb FIGURE 12.11

© 2006 by Taylor & Francis Group, LLC

325

Schematic diagram of the local charging damage during plasma etching: gate-Si etching (a) and SiO2 trench etching (b).

326

Plasma Electronics: Applications in Microelectronic Device Fabrication

plasma quasi-conductive layer

Γe

Γe jesurf Γp SiO2

jesurf charge

Si FIGURE 12.12 Local surface charges and conduction in the trench (hole) of SiO2 during etching.

12.4.8.1 Surface Continuity and Conductivity A number of experiments have emphasized that the etching of a high aspect ratio structure of SiO2 is a highly competitive process between etching of SiO2 and deposition of a thin CFx polymer, and that a thin CFx polymer is deposited on the active surface exposed to photons, neutral radicals, electrons, and ions. These facts imply that electrons may be conducted on the surface of the thin polymer layer (passivation film) under a field distribution due to a local accumulation of electrons and ions and photo-irradiation from the plasma. Then, the polymer film would prevent the wall from charging by a recombination process of the accumulated positive ions through a fast electron transport. The surface conduction in conjunction with the incident local fluxes of positive ions and electrons is described by a simple surface continuity equation (see Figure 12.12):

∂ne (r, t)

+ divΓe (r, t) = −div[je sur f (r, t)/e] − R r ne n p , ∂t sur f

∂n p

+ divΓ p (r, t) = −R r ne n p , ∂t sur f © 2006 by Taylor & Francis Group, LLC

(12.47) (12.48)

Plasma Processing and Related Topics

327

where Γe and Γ p are the instantaneous local fluxes of electrons and positive ions incident on the surface, respectively. je sur f (r, t) = σ (r, t)Esur f (r, t) is a surface current density of the electron conduction, σ is the electrical conductivity of the CFx thin polymer, and R r is the surface recombination coefficient. The time evolution of the local surface potential Vsur f (r, t) is solved at t in the system including the continuity Equations 12.47 and 12.48 and Poisson’s equation under consideration of the incident fluxes of electrons and ions from the plasma through the sheath as described in the previous section. The resulting potential distribution across the gas, surface, and solid phase is calculated by solving ∇ 2 (r, t) = −

ρs (r, t) : surface of SiO2 ε0 εr

= 0 : inside of SiO2 = −e

(12.49)

n p (r, t) − ne (r, t) : in gases, ε0

where ρs is the surface charge density on SiO2 , and ε0 and ε0 εr are the permittivities of a vacuum and SiO2 , respectively. That is, the overall potential from the gas phase to the bulk SiO2 is simultaneously solved by changing the mesh size in gas and solid. Then, the distribution of the surface charge at the boundary between SiO2 and lower poly-Si (or metal) in Figure 12.12 is iteratively calculated under the principle that the equipotential of the surface of lower poly-Si (or metal) must be maintained. The origin of the local charging of electrons and positive ions arises from the significantly different velocity distribution. In a steady state, the velocity distribution of ions incident on a wafer with a beamlike component is quite different from an isotropic distribution of electrons in the positive ion sheath, and both of the charged fluxes incident on a flat surface exposed to the plasma have the same magnitude in a time-averaged fashion. The great difference of the velocity distribution leads to a local accumulation of electrons at the upper part of the trench (hole) and ions at the lower part and bottom, that is, charging on the inside wall of a hole or trench with a high aspect ratio. Figure 12.11 demonstrates the schematic diagram of the charging damage in the gate-Si etching by Cl2 plasma (a) and SiO2 trench etching by CF4 /Ar plasma (b). Both of the chargings seriously damage the profile and lower-level device elements (i.e., the thin gate etc.).

Exercise 12.4.2 Discuss the relationship among the time constants of charging, radical deposition, and ion etching in a typical SiO2 etching (see Figure 12.13.) Typical fluxes of radicals and ions incident on a wafer are 1018 cm−2 s−1 and 1016 cm−2 s−1 , respectively. Also, the time constant for local charging is usually ms and the time for effective monolayer etching needs 100 ms.

© 2006 by Taylor & Francis Group, LLC

328

Plasma Electronics: Applications in Microelectronic Device Fabrication 1.6

Incident flux (1016 cm-2 s-1)

AR = 2.0 1.2 5.0 0.8

10.0

0.4 ion flux electron flux 0.0 0.0

1.0

2.0 3.0 Time (ms)

4.0

5.0

FIGURE 12.13 Time constant for local charging as a function of aspect ratio in typical trench etching of SiO2 .

12.4.8.2 Charging Damage to Lower Thin Elements in ULSI The current damage through thin gate oxide is typical of a high-density plasma processing (see Figure 12.11). The current densities of the direct tunnel and Fowler–Nordheim tunnel, J DT and J FNT , are expressed, respectively [20], as

J DT

1 Vox e2 V ox , φB − exp −4π dox 2em∗DT φ B − = 2 2π hdox 2 2 h 3/2 4π 2m∗FNT φ B e 3 Vox2 , J FNT = exp− 2 φ 8π hdox 3π eh B

(12.50)

(12.51)

where dox is the thickness of the oxide film, and Vox is the applied voltage between both sides of the thin oxide. h is the Planck constant, φ B is the barrier height, and m∗ is the effective mass of the electron. Plasma current damage is typical in a metal gate etching as shown in Figure 12.11a, and besides, in a multilayer interconnect system we take care of the current damage during trench or via etching of SiO2 (see Figure 12.14). 12.4.9 Thermal Damage High-energy ions are indispensable for the etching of SiO2 . There exists a highenergy deposition at a very short time within narrow spots near the material suface. The ion irradiation induces a rapid local heating and gradual cooling

© 2006 by Taylor & Francis Group, LLC

Plasma Processing and Related Topics

329

Γp

Γe

resist

V SiO2 metal I poly-Si gate oxide

LOCOS

Si

0

5 ms

10 ms

FIGURE 12.14 Current damage at the gate oxide during plasma etching.

with high-temperature gradients, leading to thermal damage such as thermal stress. The thermal damage appears on the lower-level device elements, for example, gate oxide in addition to the etching surface. In the case of the elastic binary collision at the surface, the fractional energy of the incident ion with ε p , ε =

4Mp Ms cos θ 2 ε p , (Mp + Ms )2

(12.52)

is transferred into an atom in the surface layer of the wafer. Here, θ is the incident angle of the ion to the wafer. Except for the case of Mp = Ms and θ = 0, some of the energy of the incident ion is dissipated in the form of excitation of the substrate atoms. After a short relaxation time of the highly nonequilibrium local state, a high-temperature local spot with T(r0 , t) diffuses thermally to the circumference. The thermal diffusion except for the highly nonequilibrium state during a very short time is described by ∂T ∂ T(r, t) ρc(T) = ∇r k(T) + S(r0 , t0 ) (12.53) ∂t ∂r where ρ, c(T), and k(T) are the mass density, the heat capacity, and the heat conductivity of the substrate material, respectively. S is the volume power density of the heat source, that is, the fractional ion energy dissipated to the surface. In a typical feature-scale etching by using the 2f-CCP reactor, the positive ion flux to the wafer has the magnitude on the order of 1016 cm−2 s−1 . It corresponds to the one ion impact every 1 µs. Under these conditions, the time constant of thermal diffusion is on the order of 0.1 µs to 1 µs. That is, in the micro- or nanometer scale etching, the etching by the aid of ion impact

© 2006 by Taylor & Francis Group, LLC

330

Plasma Electronics: Applications in Microelectronic Device Fabrication

is a discontinuous process in time subject to the stochastic process. Also note that when the scale of the material to be etched approaches nanometer scale, we should pay attention to the change of the physical quantity, c(T) and k(T) and so on.

References 1. Behrisch, R., Ed. 1981. Sputtering by Particle Bombardment I. Topics in Appl. Phys. 47. Berlin: Springer Verlag. 2. Behrisch, R., Ed. 1983. Sputtering by Particle Bombardment II. Topics in Appl. Phys. 52. Berlin: Springer Verlag. 3. Kress, J.D., Hansen, D.E., Voter, A.F., Liu, C.L., Liu, X.-Y., and Coronell, D.G. 1999. J. Vac. Sci. Technol. A 17:2819. 4. Engelmark, F., Westlinder, J., Nyberg, T., and Berg, S. 2003. J. Vac. Sci. Technol. A 21:1981. 5. Thompson, M.W. 1968. Philos. Mag. 18:377. 6. Kenmotsu, T., Yamamura, Y., and Ono, T. 2004. Jpn. Soc. Plasma. Sci. Nucl. Fusion Res. 7. Matsuda, A. 2004. Thin-film silicon (Inv. Review paper), Jpn. J. Appl. Phys. 43:7909. 8. Matsumura, H., Umemoto, H., and Masuda, A. 2004. J. Non-Cryst. Solids 19:338–340. 9. van Kampen, N.G. 1981. Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland. 10. Gardiner, C.W. 1983. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Berlin: Springer Verlag. 11. Ford, I.J. 1995. J. Appl. Phys. 78:510. 12. Balooch, M., Moalem, M., Wang, W-E., and Hamza, A.V. 1996. J. Vac. Sci. Technol. A 14:229. 13. Sethian, J.A. and Strain, J. 1992, 1993. J. Comp. Phys. 98: 231; Sethian, J.A. and Chopp, D.L. 1993, J. Comp. Phys. 106:77. 14. Esashi, M., and Ono, T. 2005. J. Phys. D, Topical Review 38:R223. 15. Bouchoule, A., Ed. 1999. Dusty Plasmas. Chichester: John Wiley & Sons. 16. Auciello, O. and Flamm, D.L. 1989. Plasma Diagnostics, Vol. 1. San Diego: Academic Press. 17. Bruno, G., Capezzuto, P., and Madan, A. 1995. Plasma Diagnostics of Amorphous Silicon-Based Materials. San Diego: Academic Press. 18. Saito, N., Koyama, K., and Tanimoto, M. 2003. Jpn. J. Appl. Phys. 42:Part 1, 5306. 19. Bohren, C.F., and Huffman, D.R. 1983. Absorption and Scattering of Light by Small Particles. New York: John Wiley & Sons. 20. Hirose, M. 1996. Mater. Sci. Eng. 41:35.

© 2006 by Taylor & Francis Group, LLC